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| === Sums of like powers=== | | === <code>b[n]q</code>: rebase from base b into base q === |
| ==== Sums of k m-th powers >= 0==== | | In 2005, [[User:Marc_LeBrun|Marc LeBrun]] described the rebasing notation (cf. A000695): |
| | |
| | :This may be described concisely using the "rebase" notation <code>b[n]q</code>, which means "replace b with q in the expansion of n", thus rebasing" n from base b into base q. The present sequence is <code>2[n]4</code>. Many interesting operations (e.g., <code>10[n](1/10)</code> = digit reverse, shifted) are nicely expressible this way. |
| | |
| | :Note that <code>q[n]b</code> is (roughly) inverse to <code>b[n]q</code>. |
| | |
| | :It's also natural to generalize the idea of "basis" so as to cover the likes of <code>F[n]2</code>, the so-called "fibbinary" numbers (A003714) nd provide standard ready-made images of entities obeying other arithmetics, say like <code>GF2[n]2</code> (e.g., primes = A014580, squares = the present sequence, etc.). |
| | |
| | The following table shows relevant pertinent sequences in the OEIS: |
| {| class="wikitable" style="text-align:left" | | {| class="wikitable" style="text-align:left" |
| !   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!! !! !!m=13 | | !   !!b=2!!b=3!!b=4!!b=5!!b=6!!b=7!!b=8!!b=9!!b=10 |
| |- | | |- |
| | k>=2 ||<span title="Sums of at least 2 squares s'', for s >= 4.">[https://oeis.org/A176209 A176209]</span>|| || || || || || || || || || ||  | | | q=2 ||<center>—</center>||<span title="Rebase n from 3 to 2. Replace 3^k with 2^k in ternary expansion of n.">A065361</span>||<span title="Rebase n from 4 to 2. Replace 4^k with 2^k in quaternary expansion of n.">A065362<sup>2</sup></span>||<span title="a(n)=Sum{d(i)*2^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A215088</span>||<span title="a(n)=Sum{d(i)*6^i: i=0,1,...,m}, where Sum{d(i)*2^i: i=0,1,...,m} is the base 2 representation of n.">A215089</span>||<span title="a(n) = Sum{d(i)*2^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m}=n, d(i)∈{0,1,...,6}">A203580</span>|| || ||<span title="If n = Sum c_i 10^i then a(n) = Sum c_i 2^i.">A028897</span> |
| |- | | |- |
| | k=-1 ||<span title="Sum of square displacements over all self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.">[https://oeis.org/A323856 A323856]</span>||<span title="Sum of the cubes of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294287 A294287]</span>||<span title="Sum of the 4th powers of the degrees of irreducible representations of S_n, the symmetric group on n letters.">[https://oeis.org/A129627 A129627]</span>||<span title="Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.">[https://oeis.org/A000539 A000539]</span>||<span title="Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6.">[https://oeis.org/A000540 A000540]</span>||<span title="Sum of 7th powers: 1^7 + 2^7 + ... + n^7.">[https://oeis.org/A000541 A000541]</span>||<span title="Numbers that are sums of 8th powers of 2 distinct positive integers.">[https://oeis.org/A155468 A155468]</span>||<span title="Sum of 9th powers.">[https://oeis.org/A007487 A007487]</span>||<span title="Sum of 10th powers.">[https://oeis.org/A023002 A023002]</span>|| || ||<span title="Sum of 13th powers: 0^13+1^13+2^13+...+n^13.">[https://oeis.org/A181134 A181134]</span> | | | q=3 ||<span title="Numbers n whose base 3 representation contains no 2.">A005836<sup>1</sup></span>||<center>—</center>||<span title="a(n) = Sum_{i=0..m} d(i)*3^i, where Sum_{i=0..m} d(i)*4^i is the base-4 representation of n.">A215090</span>|| ||<span title="a(n) = Sum_{i=0..m} d(i)*3^i, where Sum_{i=0..m} d(i)*6^i is the base-6 representation of n.">A215092<sup>2</sup></span>|| || || ||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 3^i.">A028898</span> |
| |- | | |- |
| | k=2 ||<span title="Sum of two squares of Lucas numbers (A000032).">[https://oeis.org/A140328 A140328]</span>||<span title="Sums of two nonnegative cubes.">[https://oeis.org/A004999 A004999]</span>||<span title="Numbers that are the sum of two 4th powers in more than one way.">[https://oeis.org/A018786 A018786]</span>|| || || || || || || || ||  | | | q=4 ||<span title="Moser-de Bruijn sequence: sums of distinct powers of 4.">A000695</span>||<span title="Numbers with no 3''s in base-4 expansion.">A023717</span>||<center>—</center>||<span title="a(n) = Sum_{i=0..m} d(i)*4^i, where Sum_{i=0..m} d(i)*5^i is the base-5 representation of n.">A303787</span>|| || || || ||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 4^i.">A028899</span> |
| |- | | |- |
| | k=3 ||<span title="Numbers that are the sum of three squares (square 0 allowed) in exactly ten ways.">[https://oeis.org/A294713 A294713]</span>||<span title="Sum of three cubes problem: a(n) = integer x with the least possible absolute value such that n = x^3 + y^3 + z^3 with |x| >= |y| >= |z|, or 0 if no such x exists.">[https://oeis.org/A332201 A332201]</span>||<span title="Numbers that are the sum of three biquadrates (fourth powers) in more than one way.">[https://oeis.org/A193244 A193244]</span>|| || || || || || || || ||  | | | q=5 ||<span title="Sums of distinct powers of 5.">A033042</span>||<span title="Positive numbers n such that the base 5 representation of n contains no 3 or 4.">A037453<sup>2</sup></span>||<span title="Sum{d(i)*5^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037459<sup>2</sup></span>||<center>—</center>||<span title="a(n) = Sum_{i=0..m} d(i)*5^i, where Sum_{i=0..m} d(i)*6^i is the base-6 representation of n.">A303788</span>|| || || ||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 5^i.">A028900</span> |
| |- | | |- |
| | k=4 || ||<span title="Numbers that are the sum of 4 cubes in more than 1 way.">[https://oeis.org/A001245 A001245]</span>|| || || || || || || || || ||  | | | q=6 ||<span title="Sums of distinct powers of 6.">A033043</span>||<span title="a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*3^i is the base 3 representation of n.">A037454</span>||<span title="a(n)=Sum{d(i)*6^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037460</span>||<span title="Sum{d(i)*6^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037465</span>||<center>—</center>||<span title="a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*7^i is the base-7 representation of n.">A303789</span>|| || ||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 6^i.">A028901</span> |
| |- | | |- |
| |} | | | q=7 ||<span title="Sums of distinct powers of 7.">A033044<sup>1</sup></span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*3^i: i=0,1,...,m} is the base 3 representation of n.">A037455</span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037461<sup>2</sup></span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037466</span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.">A037470</span>||<center>—</center>|| || ||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 7^i.">A028902</span> |
| ==== Sums of exactly k positive m-th powers > 0====
| |
| {| class="wikitable" style="text-align:left"
| |
| !   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
| |
| |-
| |
| | k=2 ||<span title="Numbers that are the sum of 2 nonzero squares, including repetitions.">[https://oeis.org/A024509 A024509]</span>||<span title="Numbers that are the sum of 2 positive cubes.">[https://oeis.org/A003325 A003325]</span>||<span title="Numbers that are the sum of 2 positive 4th powers.">[https://oeis.org/A003336 A003336]</span>||<span title="Numbers that are the sum of 2 positive 5th powers.">[https://oeis.org/A003347 A003347]</span>||<span title="Numbers that are the sum of 2 nonzero 6th powers.">[https://oeis.org/A003358 A003358]</span>||<span title="Numbers that are the sum of 2 positive 7th powers.">[https://oeis.org/A003369 A003369]</span>||<span title="Numbers that are the sum of 2 nonzero 8th powers.">[https://oeis.org/A003380 A003380]</span>||<span title="Numbers that are the sum of 2 positive 9th powers.">[https://oeis.org/A003391 A003391]</span>||<span title="Numbers that are the sum of 2 nonzero 10th powers.">[https://oeis.org/A004802 A004802]</span>||<span title="Numbers that are the sum of 2 positive 11th powers.">[https://oeis.org/A004813 A004813]</span>
| |
| |-
| |
| | k=3 ||<span title="Numbers that are the sum of 3 nonzero squares, including repetitions.">[https://oeis.org/A024795 A024795]</span>||<span title="Numbers that are the sum of 3 positive cubes, including repetitions.">[https://oeis.org/A024981 A024981]</span>||<span title="Numbers that are the sum of 3 nonzero 4th powers in more than one way.">[https://oeis.org/A309762 A309762]</span>||<span title="Numbers that are the sum of 3 positive 5th powers.">[https://oeis.org/A003348 A003348]</span>||<span title="Numbers that are the sum of 3 nonzero 6th powers.">[https://oeis.org/A003359 A003359]</span>||<span title="Numbers that are the sum of 3 positive 7th powers.">[https://oeis.org/A003370 A003370]</span>||<span title="Numbers that are the sum of 3 nonzero 8th powers.">[https://oeis.org/A003381 A003381]</span>||<span title="Numbers that are the sum of 3 positive 9th powers.">[https://oeis.org/A003392 A003392]</span>||<span title="Numbers that are the sum of 3 nonzero 10th powers.">[https://oeis.org/A004803 A004803]</span>||<span title="Numbers that are the sum of 3 positive 11th powers.">[https://oeis.org/A004814 A004814]</span>
| |
| |-
| |
| | k=4 ||<span title="Numbers that are the sum of 4 nonzero squares.">[https://oeis.org/A000414 A000414]</span>|| ||<span title="Numbers that are the sum of 4 nonzero 4th powers in more than one way.">[https://oeis.org/A309763 A309763]</span>||<span title="Numbers that are the sum of 4 positive 5th powers.">[https://oeis.org/A003349 A003349]</span>||<span title="Numbers that are the sum of 4 positive 6th powers.">[https://oeis.org/A003360 A003360]</span>||<span title="Numbers that are the sum of 4 positive 7th powers.">[https://oeis.org/A003371 A003371]</span>||<span title="Numbers that are the sum of 4 nonzero 8th powers.">[https://oeis.org/A003382 A003382]</span>||<span title="Numbers that are the sum of 4 positive 9th powers.">[https://oeis.org/A003393 A003393]</span>||<span title="Numbers that are the sum of 4 nonzero 10th powers.">[https://oeis.org/A004804 A004804]</span>||<span title="Numbers that are the sum of 4 positive 11th powers.">[https://oeis.org/A004815 A004815]</span>
| |
| |-
| |
| | k=5 ||<span title="Numbers that are the sum of 5 positive squares.">[https://oeis.org/A047700 A047700]</span>||<span title="Numbers that are the sum of 5 positive cubes.">[https://oeis.org/A003328 A003328]</span>||<span title="Numbers that are the sum of 5 positive 4th powers.">[https://oeis.org/A003339 A003339]</span>||<span title="Numbers that are the sum of 5 positive 5th powers.">[https://oeis.org/A003350 A003350]</span>||<span title="Numbers that are the sum of 5 positive 6th powers.">[https://oeis.org/A003361 A003361]</span>||<span title="Numbers that are the sum of 5 positive 7th powers.">[https://oeis.org/A003372 A003372]</span>||<span title="Numbers that are the sum of 5 nonzero 8th powers.">[https://oeis.org/A003383 A003383]</span>||<span title="Numbers that are the sum of 5 positive 9th powers.">[https://oeis.org/A003394 A003394]</span>||<span title="Numbers that are the sum of 5 positive 10th powers.">[https://oeis.org/A004805 A004805]</span>||<span title="Numbers that are the sum of 5 positive 11th powers.">[https://oeis.org/A004816 A004816]</span>
| |
| |-
| |
| | k=6 || ||<span title="Numbers that are the sum of 6 positive cubes.">[https://oeis.org/A003329 A003329]</span>||<span title="Numbers that are the sum of 6 positive 4th powers.">[https://oeis.org/A003340 A003340]</span>||<span title="Numbers that are the sum of 6 positive 5th powers.">[https://oeis.org/A003351 A003351]</span>||<span title="Numbers that are the sum of 6 positive 6th powers.">[https://oeis.org/A003362 A003362]</span>||<span title="Numbers that are the sum of 6 positive 7th powers.">[https://oeis.org/A003373 A003373]</span>||<span title="Numbers that are the sum of 6 nonzero 8th powers.">[https://oeis.org/A003384 A003384]</span>||<span title="Numbers that are the sum of 6 positive 9th powers.">[https://oeis.org/A003395 A003395]</span>||<span title="Numbers that are the sum of 6 positive 10th powers.">[https://oeis.org/A004806 A004806]</span>||<span title="Numbers that are the sum of 6 positive 11th powers.">[https://oeis.org/A004817 A004817]</span>
| |
| |-
| |
| | k=7 || ||<span title="Numbers that are the sum of 7 positive cubes.">[https://oeis.org/A003330 A003330]</span>||<span title="Numbers that are the sum of 7 positive 4th powers.">[https://oeis.org/A003341 A003341]</span>||<span title="Numbers that are the sum of 7 positive 5th powers.">[https://oeis.org/A003352 A003352]</span>||<span title="Numbers that are the sum of 7 positive 6th powers.">[https://oeis.org/A003363 A003363]</span>||<span title="Numbers that are the sum of 7 positive 7th powers.">[https://oeis.org/A003374 A003374]</span>||<span title="Numbers that are the sum of 7 nonzero 8th powers.">[https://oeis.org/A003385 A003385]</span>||<span title="Numbers that are the sum of 7 positive 9th powers.">[https://oeis.org/A003396 A003396]</span>||<span title="Numbers that are the sum of 7 positive 10th powers.">[https://oeis.org/A004807 A004807]</span>||<span title="Numbers that are the sum of 7 positive 11th powers.">[https://oeis.org/A004818 A004818]</span>
| |
| |-
| |
| | k=8 || ||<span title="Numbers that are the sum of 8 positive cubes.">[https://oeis.org/A003331 A003331]</span>||<span title="Numbers that are the sum of 8 positive 4th powers.">[https://oeis.org/A003342 A003342]</span>||<span title="Numbers that are the sum of 8 positive 5th powers.">[https://oeis.org/A003353 A003353]</span>||<span title="Numbers that are the sum of 8 positive 6th powers.">[https://oeis.org/A003364 A003364]</span>||<span title="Numbers that are the sum of 8 positive 7th powers.">[https://oeis.org/A003375 A003375]</span>||<span title="Numbers that are the sum of 8 nonzero 8th powers.">[https://oeis.org/A003386 A003386]</span>||<span title="Numbers that are the sum of 8 positive 9th powers.">[https://oeis.org/A003397 A003397]</span>||<span title="Numbers that are the sum of 8 positive 10th powers.">[https://oeis.org/A004808 A004808]</span>||<span title="Numbers that are the sum of 8 positive 11th powers.">[https://oeis.org/A004819 A004819]</span>
| |
| |-
| |
| | k=9 || ||<span title="Numbers that are the sum of 9 positive cubes.">[https://oeis.org/A003332 A003332]</span>||<span title="Numbers that are the sum of 9 positive 4th powers.">[https://oeis.org/A003343 A003343]</span>||<span title="Numbers that are the sum of 9 positive 5th powers.">[https://oeis.org/A003354 A003354]</span>||<span title="Numbers that are the sum of 9 positive 6th powers.">[https://oeis.org/A003365 A003365]</span>||<span title="Numbers that are the sum of 9 positive 7th powers.">[https://oeis.org/A003376 A003376]</span>||<span title="Numbers that are the sum of 9 nonzero 8th powers.">[https://oeis.org/A003387 A003387]</span>||<span title="Numbers that are the sum of 9 positive 9th powers.">[https://oeis.org/A003398 A003398]</span>||<span title="Numbers that are the sum of 9 positive 10th powers.">[https://oeis.org/A004809 A004809]</span>||<span title="Numbers that are the sum of 9 positive 11th powers.">[https://oeis.org/A004820 A004820]</span>
| |
| |- | | |- |
| | k=10 || ||<span title="Numbers that are the sum of 10 positive cubes.">[https://oeis.org/A003333 A003333]</span>||<span title="Numbers that are the sum of 10 positive 4th powers.">[https://oeis.org/A003344 A003344]</span>||<span title="Numbers that are the sum of 10 positive 5th powers.">[https://oeis.org/A003355 A003355]</span>||<span title="Numbers that are the sum of 10 positive 6th powers.">[https://oeis.org/A003366 A003366]</span>||<span title="Numbers that are the sum of 10 positive 7th powers.">[https://oeis.org/A003377 A003377]</span>||<span title="Sum of 10 nonzero 8th powers.">[https://oeis.org/A003388 A003388]</span>||<span title="Sum of 10 positive 9th powers.">[https://oeis.org/A003399 A003399]</span>||<span title="Numbers that are the sum of 10 positive 10th powers.">[https://oeis.org/A004810 A004810]</span>||<span title="Numbers that are the sum of 10 positive 11th powers.">[https://oeis.org/A004821 A004821]</span> | | | q=8 ||<span title="Sums of distinct powers of 8.">A033045</span>||<span title="a(n)=Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*3^i: i=0,1,...,m} is the base 3 representation of n.">A037456</span>||<span title="a(n) = Sum_{i = 0..m} d(i)*8^i, where Sum_{i = 0..m} d(i)*4^i is the base 4 representation of n.">A037462</span>||<span title="Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037467</span>||<span title="a(n)=Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.">A037471</span>||<span title="a(n) = Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.">A037474</span>||<center>—</center>|| ||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 8^i.">A028903</span> |
| |- | | |- |
| | k=11 || ||<span title="Numbers that are the sum of 11 positive cubes.">[https://oeis.org/A003334 A003334]</span>||<span title="Numbers that are the sum of 11 positive 4th powers.">[https://oeis.org/A003345 A003345]</span>||<span title="Numbers that are the sum of 11 positive 5th powers.">[https://oeis.org/A003356 A003356]</span>||<span title="Numbers that are the sum of 11 positive 6th powers.">[https://oeis.org/A003367 A003367]</span>||<span title="Numbers that are the sum of 11 positive 7th powers.">[https://oeis.org/A003378 A003378]</span>||<span title="Numbers that are the sum of 11 positive 8th powers.">[https://oeis.org/A003389 A003389]</span>||<span title="Sum of 11 positive 9th powers.">[https://oeis.org/A004800 A004800]</span>||<span title="Numbers that are the sum of 11 positive 10th powers.">[https://oeis.org/A004811 A004811]</span>||<span title="Numbers that are the sum of 11 positive 11th powers.">[https://oeis.org/A004822 A004822]</span> | | | q=9 ||<span title="Sums of distinct powers of 9.">A033046</span>|| ||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037463</span>||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037468</span>||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.">A037472</span>||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.">A037475</span>||<span title="a(n) = Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*8^i: i=0,1,...,m} is the base 8 representation of n.">A037477</span>||<center>—</center>||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 9^i.">A028904</span> |
| |- | | |- |
| | k=12 || ||<span title="Numbers that are the sum of 12 positive cubes.">[https://oeis.org/A003335 A003335]</span>||<span title="Numbers that are the sum of 12 positive 4th powers.">[https://oeis.org/A003346 A003346]</span>||<span title="Numbers that are the sum of 12 positive 5th powers.">[https://oeis.org/A003357 A003357]</span>||<span title="Numbers that are the sum of 12 positive 6th powers.">[https://oeis.org/A003368 A003368]</span>||<span title="Numbers that are the sum of 12 positive 7th powers.">[https://oeis.org/A003379 A003379]</span>||<span title="Sum of 12 nonzero 8th powers.">[https://oeis.org/A003390 A003390]</span>||<span title="Sum of 12 positive 9th powers.">[https://oeis.org/A004801 A004801]</span>||<span title="Numbers that are the sum of 12 positive 10th powers.">[https://oeis.org/A004812 A004812]</span>||<span title="Numbers that are the sum of 12 positive 11th powers.">[https://oeis.org/A004823 A004823]</span> | | | q=10 ||<span title="The binary numbers (or binary words, or binary vectors): numbers written in base 2.">A007088</span>||<span title="Numbers in base 3.">A007089</span>||<span title="Numbers in base 4.">A007090</span>||<span title="Numbers in base 5.">A007091</span>||<span title="Numbers in base 6.">A007092</span>||<span title="Numbers in base 7.">A007093</span>||<span title="Numbers in base 8.">A007094</span>||<span title=" Numbers in base 9.">A007095</span><br><span title="a(n)=Sum{d(i)*10^i: i=0,1,...,m}, where Sum{d(i)*9^i: i=0,1,...,m} is the base 9 representation of n.">A037479<sup>2</sup></span>||<center>—</center> |
| |-
| |
| | k=13 || || || ||<span title="Sum of 13 positive 5th powers.">[https://oeis.org/A123294 A123294]</span>|| || || || || || 
| |
| |-
| |
| | k=14 || || || ||<span title="Sum of 14 positive 5th powers.">[https://oeis.org/A123295 A123295]</span>|| || || || || || 
| |
| |-
| |
| |}
| |
| ==== Sums of at most k positive m-th powers > 0====
| |
| {| class="wikitable" style="text-align:left"
| |
| !   !!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
| |
| |-
| |
| | k<=2 || ||<span title="Numbers that are the sum of at most 2 nonzero 4th powers.">[https://oeis.org/A004831 A004831]</span>||<span title="Numbers that are the sum of at most 2 positive 5th powers.">[https://oeis.org/A004842 A004842]</span>||<span title="Numbers that are the sum of at most 2 nonzero 6th powers.">[https://oeis.org/A004853 A004853]</span>||<span title="Numbers that are the sum of at most 2 positive 7th powers.">[https://oeis.org/A004864 A004864]</span>||<span title="Numbers that are the sum of at most 2 nonzero 8th powers.">[https://oeis.org/A004875 A004875]</span>||<span title="Numbers that are the sum of at most 2 positive 9th powers.">[https://oeis.org/A004886 A004886]</span>||<span title="Numbers that are the sum of at most 2 nonzero 10th powers.">[https://oeis.org/A004897 A004897]</span>||<span title="Numbers that are the sum of at most 2 positive 11th powers.">[https://oeis.org/A004908 A004908]</span>
| |
| |-
| |
| | k<=3 ||<span title="Numbers that are the sum of at most 3 positive cubes.">[https://oeis.org/A004825 A004825]</span>||<span title="Numbers that are the sum of at most 3 nonzero 4th powers.">[https://oeis.org/A004832 A004832]</span>||<span title="Numbers that are the sum of at most 3 positive 5th powers.">[https://oeis.org/A004843 A004843]</span>||<span title="Numbers that are the sum of at most 3 nonzero 6th powers.">[https://oeis.org/A004854 A004854]</span>||<span title="Numbers that are the sum of at most 3 positive 7th powers.">[https://oeis.org/A004865 A004865]</span>||<span title="Numbers that are the sum of at most 3 nonzero 8th powers.">[https://oeis.org/A004876 A004876]</span>||<span title="Numbers that are the sum of at most 3 positive 9th powers.">[https://oeis.org/A004887 A004887]</span>||<span title="Numbers that are the sum of at most 3 nonzero 10th powers.">[https://oeis.org/A004898 A004898]</span>||<span title="Numbers that are the sum of at most 3 positive 11th powers.">[https://oeis.org/A004909 A004909]</span>
| |
| |-
| |
| | k<=4 ||<span title="Numbers that are the sum of at most 4 positive cubes.">[https://oeis.org/A004826 A004826]</span>||<span title="Numbers that are the sum of at most 4 nonzero 4th powers.">[https://oeis.org/A004833 A004833]</span>||<span title="Numbers that are the sum of at most 4 positive 5th powers.">[https://oeis.org/A004844 A004844]</span>||<span title="Numbers that are the sum of at most 4 nonzero 6th powers.">[https://oeis.org/A004855 A004855]</span>||<span title="Numbers that are the sum of at most 4 positive 7th powers.">[https://oeis.org/A004866 A004866]</span>||<span title="Numbers that are the sum of at most 4 nonzero 8th powers.">[https://oeis.org/A004877 A004877]</span>||<span title="Numbers that are the sum of at most 4 positive 9th powers.">[https://oeis.org/A004888 A004888]</span>||<span title="Numbers that are the sum of at most 4 nonzero 10th powers.">[https://oeis.org/A004899 A004899]</span>||<span title="Numbers that are the sum of at most 4 positive 11th powers.">[https://oeis.org/A004910 A004910]</span>
| |
| |-
| |
| | k<=5 ||<span title="Numbers that are the sum of at most 5 positive cubes.">[https://oeis.org/A004827 A004827]</span>||<span title="Numbers that are the sum of at most 5 nonzero 4th powers.">[https://oeis.org/A004834 A004834]</span>||<span title="Numbers that are the sum of at most 5 positive 5th powers.">[https://oeis.org/A004845 A004845]</span>||<span title="Numbers that are the sum of at most 5 nonzero 6th powers.">[https://oeis.org/A004856 A004856]</span>||<span title="Numbers that are the sum of at most 5 positive 7th powers.">[https://oeis.org/A004867 A004867]</span>||<span title="Numbers that are the sum of at most 5 nonzero 8th powers.">[https://oeis.org/A004878 A004878]</span>||<span title="Numbers that are the sum of at most 5 positive 9th powers.">[https://oeis.org/A004889 A004889]</span>||<span title="Numbers that are the sum of at most 5 nonzero 10th powers.">[https://oeis.org/A004900 A004900]</span>||<span title="Numbers that are the sum of at most 5 positive 11th powers.">[https://oeis.org/A004911 A004911]</span>
| |
| |-
| |
| | k<=6 ||<span title="Numbers that are the sum of at most 6 positive cubes.">[https://oeis.org/A004828 A004828]</span>||<span title="Numbers that are the sum of at most 6 nonzero 4th powers.">[https://oeis.org/A004835 A004835]</span>||<span title="Numbers that are the sum of at most 6 positive 5th powers.">[https://oeis.org/A004846 A004846]</span>||<span title="Numbers that are the sum of at most 6 nonzero 6th powers.">[https://oeis.org/A004857 A004857]</span>||<span title="Numbers that are the sum of at most 6 positive 7th powers.">[https://oeis.org/A004868 A004868]</span>||<span title="Numbers that are the sum of at most 6 nonzero 8th powers.">[https://oeis.org/A004879 A004879]</span>||<span title="Numbers that are the sum of at most 6 positive 9th powers.">[https://oeis.org/A004890 A004890]</span>||<span title="Numbers that are the sum of at most 6 nonzero 10th powers.">[https://oeis.org/A004901 A004901]</span>||<span title="Numbers that are the sum of at most 6 positive 11th powers.">[https://oeis.org/A004912 A004912]</span>
| |
| |-
| |
| | k<=7 ||<span title="Numbers that are the sum of at most 7 positive cubes.">[https://oeis.org/A004829 A004829]</span>||<span title="Numbers that are the sum of at most 7 nonzero 4th powers.">[https://oeis.org/A004836 A004836]</span>||<span title="Numbers that are the sum of at most 7 positive 5th powers.">[https://oeis.org/A004847 A004847]</span>||<span title="Numbers that are the sum of at most 7 nonzero 6th powers.">[https://oeis.org/A004858 A004858]</span>||<span title="Numbers that are the sum of at most 7 positive 7th powers.">[https://oeis.org/A004869 A004869]</span>||<span title="Numbers that are the sum of at most 7 nonzero 8th powers.">[https://oeis.org/A004880 A004880]</span>||<span title="Numbers that are the sum of at most 7 positive 9th powers.">[https://oeis.org/A004891 A004891]</span>||<span title="Numbers that are the sum of at most 7 nonzero 10th powers.">[https://oeis.org/A004902 A004902]</span>||<span title="Numbers that are the sum of at most 7 positive 11th powers.">[https://oeis.org/A004913 A004913]</span>
| |
| |-
| |
| | k<=8 ||<span title="Numbers that are the sum of at most 8 positive cubes.">[https://oeis.org/A004830 A004830]</span>||<span title="Numbers that are the sum of at most 8 nonzero 4th powers.">[https://oeis.org/A004837 A004837]</span>||<span title="Numbers that are the sum of at most 8 positive 5th powers.">[https://oeis.org/A004848 A004848]</span>||<span title="Numbers that are the sum of at most 8 nonzero 6th powers.">[https://oeis.org/A004859 A004859]</span>||<span title="Numbers that are the sum of at most 8 positive 7th powers.">[https://oeis.org/A004870 A004870]</span>||<span title="Numbers that are the sum of at most 8 nonzero 8th powers.">[https://oeis.org/A004881 A004881]</span>||<span title="Numbers that are the sum of at most 8 positive 9th powers.">[https://oeis.org/A004892 A004892]</span>||<span title="Numbers that are the sum of at most 8 nonzero 10th powers.">[https://oeis.org/A004903 A004903]</span>||<span title="Numbers that are the sum of at most 8 positive 11th powers.">[https://oeis.org/A004914 A004914]</span>
| |
| |-
| |
| | k<=9 || ||<span title="Numbers that are the sum of at most 9 nonzero 4th powers.">[https://oeis.org/A004838 A004838]</span>||<span title="Numbers that are the sum of at most 9 positive 5th powers.">[https://oeis.org/A004849 A004849]</span>||<span title="Numbers that are the sum of at most 9 nonzero 6th powers.">[https://oeis.org/A004860 A004860]</span>||<span title="Numbers that are the sum of at most 9 positive 7th powers.">[https://oeis.org/A004871 A004871]</span>||<span title="Numbers that are the sum of at most 9 nonzero 8th powers.">[https://oeis.org/A004882 A004882]</span>||<span title="Numbers that are the sum of at most 9 positive 9th powers.">[https://oeis.org/A004893 A004893]</span>||<span title="Numbers that are the sum of at most 9 nonzero 10th powers.">[https://oeis.org/A004904 A004904]</span>||<span title="Numbers that are the sum of at most 9 positive 11th powers.">[https://oeis.org/A004915 A004915]</span>
| |
| |-
| |
| | k<=10 || ||<span title="Numbers that are the sum of at most 10 nonzero 4th powers.">[https://oeis.org/A004839 A004839]</span>||<span title="Numbers that are the sum of at most 10 positive 5th powers.">[https://oeis.org/A004850 A004850]</span>||<span title="Numbers that are the sum of at most 10 nonzero 6th powers.">[https://oeis.org/A004861 A004861]</span>||<span title="Numbers that are the sum of at most 10 positive 7th powers.">[https://oeis.org/A004872 A004872]</span>||<span title="Numbers that are the sum of at most 10 nonzero 8th powers.">[https://oeis.org/A004883 A004883]</span>||<span title="Numbers that are the sum of at most 10 positive 9th powers.">[https://oeis.org/A004894 A004894]</span>||<span title="Numbers that are the sum of at most 10 nonzero 10th powers.">[https://oeis.org/A004905 A004905]</span>||<span title="Numbers that are the sum of at most 10 positive 11th powers.">[https://oeis.org/A004916 A004916]</span>
| |
| |-
| |
| | k<=11 || ||<span title="Numbers that are the sum of at most 11 nonzero 4th powers.">[https://oeis.org/A004840 A004840]</span>||<span title="Numbers that are the sum of at most 11 positive 5th powers.">[https://oeis.org/A004851 A004851]</span>||<span title="Numbers that are the sum of at most 11 nonzero 6th powers.">[https://oeis.org/A004862 A004862]</span>||<span title="Numbers that are the sum of at most 11 positive 7th powers.">[https://oeis.org/A004873 A004873]</span>||<span title="Numbers that are the sum of at most 11 nonzero 8th powers.">[https://oeis.org/A004884 A004884]</span>||<span title="Numbers that are the sum of at most 11 positive 9th powers.">[https://oeis.org/A004895 A004895]</span>||<span title="Numbers that are the sum of at most 11 nonzero 10th powers.">[https://oeis.org/A004906 A004906]</span>||<span title="Numbers that are the sum of at most 11 positive 11th powers.">[https://oeis.org/A004917 A004917]</span>
| |
| |-
| |
| | k<=12 || ||<span title="Numbers that are the sum of at most 12 nonzero 4th powers.">[https://oeis.org/A004841 A004841]</span>||<span title="Numbers that are the sum of at most 12 positive 5th powers.">[https://oeis.org/A004852 A004852]</span>||<span title="Numbers that are the sum of at most 12 nonzero 6th powers.">[https://oeis.org/A004863 A004863]</span>||<span title="Numbers that are the sum of at most 12 positive 7th powers.">[https://oeis.org/A004874 A004874]</span>||<span title="Numbers that are the sum of at most 12 nonzero 8th powers.">[https://oeis.org/A004885 A004885]</span>||<span title="Numbers that are the sum of at most 12 positive 9th powers.">[https://oeis.org/A004896 A004896]</span>||<span title="Numbers that are the sum of at most 12 nonzero 10th powers.">[https://oeis.org/A004907 A004907]</span>||<span title="Numbers that are the sum of at most 12 positive 11th powers.">[https://oeis.org/A004918 A004918]</span>
| |
| |-
| |
| |}
| |
| ==== Sums of k positive m-th powers > 1====
| |
| {| class="wikitable" style="text-align:left"
| |
| !   !!m=2!!m=3
| |
| |-
| |
| | k=-1 || ||<span title="Numbers which can be written as sum of cubes > 1.">[https://oeis.org/A078131 A078131]</span>
| |
| |-
| |
| | k=2 || ||<span title="Numbers that are the sum of 2 cubes > 1.">[https://oeis.org/A294073 A294073]</span>
| |
| |-
| |
| | k=3 ||<span title="Numbers that are the sum of 3 squares > 1.">[https://oeis.org/A302359 A302359]</span>||<span title="Numbers that are the sum of 3 cubes > 1.">[https://oeis.org/A302360 A302360]</span> | |
| |-
| |
| |}
| |
| ==== Numbers n having exactly k representations as the sum of m squares >= 0====
| |
| <!--A295158 quant_eq ten representations as the sum of five least_0 pow_2. -->
| |
| {| class="wikitable" style="text-align:left"
| |
| !   !!m=2!! !! !!m=5!!m=6!!m=7
| |
| |- | | |- |
| | k=1 || || || || ||<span title="Numbers that have exactly one representation as a sum of six nonnegative squares.">[https://oeis.org/A295484 A295484]</span>||  | | | q=11 ||<span title="Sums of distinct powers of 11.">A033047</span>|| || || || || || || ||  |
| |- | | |- |
| | k=2 ||<span title="Numbers that are the sum of 2 squares in exactly 2 ways.">[https://oeis.org/A085625 A085625]</span>|| || ||<span title="Numbers that have exactly two representations as a sum of five nonnegative squares.">[https://oeis.org/A295150 A295150]</span>||<span title="Numbers that have exactly two representations as a sum of six nonnegative squares.">[https://oeis.org/A295485 A295485]</span>||<span title="Numbers that have exactly two representations of a sum of seven nonnegative squares.">[https://oeis.org/A295742 A295742]</span> | | | q=12 ||<span title="Sums of distinct powers of 12.">A033048</span>|| || || || || || || ||<span title="Numbers in base-12 representation that can be written with decimal digits.">A102487<sup>1</sup></span> |
| |- | | |- |
| | k=3 ||<span title="Numbers that are the sum of 2 squares in exactly 3 ways.">[https://oeis.org/A000443 A000443]</span>|| || ||<span title="Numbers that have exactly three representations as a sum of five nonnegative squares.">[https://oeis.org/A295151 A295151]</span>||<span title="Numbers that have exactly three representations as a sum of six nonnegative squares.">[https://oeis.org/A295486 A295486]</span>||<span title="Numbers that have exactly three representations of a sum of seven nonnegative squares.">[https://oeis.org/A295743 A295743]</span> | | | q=13 ||<span title="Sums of distinct powers of 13.">A033049</span>|| || || || || || || ||<span title="If n = c0 + c1*10 + c2*10^2 + ...cn*10^n then a(n) = c0 + c1*13 + c2*13^2 + ...cn*13^k.">A094823</span> |
| |- | | |- |
| | k=4 || || || ||<span title="Numbers that have exactly four representations as a sum of five nonnegative squares.">[https://oeis.org/A295152 A295152]</span>||<span title="Numbers that have exactly four representations as a sum of six nonnegative squares.">[https://oeis.org/A295487 A295487]</span>||<span title="Numbers that have exactly four representations of a sum of seven nonnegative squares.">[https://oeis.org/A295744 A295744]</span> | | | q=14 ||<span title="Numbers whose set of base 14 digits is {0,1}.">A033050</span>|| || || || || || || ||  |
| |- | | |- |
| | k=5 ||<span title="Numbers that are the sum of 2 squares in exactly 5 ways.">[https://oeis.org/A294716 A294716]</span>|| || ||<span title="Numbers that have exactly five representations as a sum of five nonnegative squares.">[https://oeis.org/A295153 A295153]</span>||<span title="Numbers that have exactly five representations as a sum of six nonnegative squares.">[https://oeis.org/A295488 A295488]</span>||<span title="Numbers that have exactly five representations of a sum of seven nonnegative squares.">[https://oeis.org/A295745 A295745]</span> | | | q=15 ||<span title="Numbers whose set of base 15 digits is {0,1}.">A033051</span>|| || || || || || || ||  |
| |- | | |- |
| | k=6 || || || ||<span title="Numbers that have exactly six representations as a sum of five nonnegative squares.">[https://oeis.org/A295154 A295154]</span>||<span title="Numbers that have exactly six representations as a sum of six nonnegative squares.">[https://oeis.org/A295489 A295489]</span>||<span title="Numbers that have exactly six representations of a sum of seven nonnegative squares.">[https://oeis.org/A295747 A295747]</span> | | | q=16 ||<span title="a(1) = 1, a(2n) = 16a(n), a(2n+1) = a(2n)+1.">A033052</span>|| || || || || || || ||<span title="Take the decimal representation of n and read it as if it were written in hexadecimal.">A102489</span> |
| |- | | |- |
| | k=7 || || || ||<span title="Numbers that have exactly seven representations as a sum of five nonnegative squares.">[https://oeis.org/A295155 A295155]</span>||<span title="Numbers that have exactly seven representations as a sum of six nonnegative squares.">[https://oeis.org/A295490 A295490]</span>||<span title="Numbers that have exactly seven representations of a sum of seven nonnegative squares.">[https://oeis.org/A295748 A295748]</span> | | | q=17 ||<span title="a(0)=0, a(1)=1, a(2n)=17*a(n), a(2n+1)=a(2n)+1.">A197351</span>|| || || || || || || ||  |
| |- | | |- |
| | k=8 || || || ||<span title="Numbers that have exactly eight representations as a sum of five nonnegative squares.">[https://oeis.org/A295156 A295156]</span>||<span title="Numbers that have exactly eight representations as a sum of six nonnegative squares.">[https://oeis.org/A295491 A295491]</span>||<span title="Numbers that have exactly eight representations of a sum of seven nonnegative squares.">[https://oeis.org/A295749 A295749]</span> | | | q=18 ||<span title="a(0)=0, a(1)=1, a(2n)=18*a(n), a(2n+1)=a(2n)+1.">A197352</span>|| || || || || || || ||  |
| |- | | |- |
| | k=9 || || || ||<span title="Numbers that have exactly nine representations as a sum of five nonnegative squares.">[https://oeis.org/A295157 A295157]</span>||<span title="Numbers that have exactly nine representations as a sum of six nonnegative squares.">[https://oeis.org/A295492 A295492]</span>||<span title="Numbers that have exactly nine representations of a sum of seven nonnegative squares.">[https://oeis.org/A295750 A295750]</span> | | | q=19 ||<span title="a(0)=0, a(1)=1, a(2n)=19*a(n), a(2n+1)=a(2n)+1.">A197353</span>|| || || || || || || ||  |
| |- | | |- |
| | k=10 || || || ||<span title="Numbers that have exactly ten representations as a sum of five nonnegative squares.">[https://oeis.org/A295158 A295158]</span>||<span title="Numbers that have exactly ten representations as a sum of six nonnegative squares.">[https://oeis.org/A295493 A295493]</span>||<span title="Numbers that have exactly ten representations as a sum of seven nonnegative squares.">[https://oeis.org/A295751 A295751]</span> | | | q=20 ||<span title="Sum of distinct powers of 20; i.e., numbers with digits in {0,1} base 20; i.e., write n in base 2 and read as if written in base 20.">A063012</span>|| || || || || || || ||<span title="Numbers whose base-20 representation can be written with decimal digits.">A102491<sup>1</sup></span> |
| |- | | |- |
| |} | | |} |
| | <sup>1</sup> These sequences have offset 1 and start with n=0.<br> |
| | <sup>2</sup> These sequences have offset 1 and start with n=1.<br> |
| | All other sequences have offset 0 and start with n=0.<br> |
| | ===Sums of distinct powers of q=== |
| | The first column (b=2) of the table above shows the sequences for ''Sums of distinct powers of q'', since the binary digits in n enumerate all such powers. |
| | ===Examples=== |
| | A037454: 3[n]6 |
| | n = 0 1 2 3 4 5 6 7 8 9 10 11 |
| | a(n) = 0, 1, 2, 6, 7, 8, 12, 13, 14, 36, 37, 38, ... |
| | n = 11: 11<sub>10</sub> = 102<sub>3</sub> -> 102<sub>6</sub> = 1*6^2 + 0*6^1 + 2*6^0 = 38<sub>10</sub> = a(11) |
| | ===Programs=== |
| | * (Mathematica) |
| | b:=3; q:=6; Table[FromDigits[RealDigits[n, b], q], {n, 0, 100}] |
| | * (PARI) |
| | b=3; q=6; for(n=0,100,print1(fromdigits(digits(n, b), q),",")); |
| | * Java ([https://github.com/archmageirvine/joeis jOEIS]) |
| | java -cp joeis.jar [https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a037/A037454.java irvine.oeis.a037.A037454] 3 6 |
b[n]q: rebase from base b into base q
In 2005, Marc LeBrun described the rebasing notation (cf. A000695):
- This may be described concisely using the "rebase" notation
b[n]q, which means "replace b with q in the expansion of n", thus rebasing" n from base b into base q. The present sequence is 2[n]4. Many interesting operations (e.g., 10[n](1/10) = digit reverse, shifted) are nicely expressible this way.
- Note that
q[n]b is (roughly) inverse to b[n]q.
- It's also natural to generalize the idea of "basis" so as to cover the likes of
F[n]2, the so-called "fibbinary" numbers (A003714) nd provide standard ready-made images of entities obeying other arithmetics, say like GF2[n]2 (e.g., primes = A014580, squares = the present sequence, etc.).
The following table shows relevant pertinent sequences in the OEIS:
| |
b=2 |
b=3 |
b=4 |
b=5 |
b=6 |
b=7 |
b=8 |
b=9 |
b=10
|
| q=2 |
— |
A065361 |
A0653622 |
A215088 |
A215089 |
A203580 |
|
|
A028897
|
| q=3 |
A0058361 |
— |
A215090 |
|
A2150922 |
|
|
|
A028898
|
| q=4 |
A000695 |
A023717 |
— |
A303787 |
|
|
|
|
A028899
|
| q=5 |
A033042 |
A0374532 |
A0374592 |
— |
A303788 |
|
|
|
A028900
|
| q=6 |
A033043 |
A037454 |
A037460 |
A037465 |
— |
A303789 |
|
|
A028901
|
| q=7 |
A0330441 |
A037455 |
A0374612 |
A037466 |
A037470 |
— |
|
|
A028902
|
| q=8 |
A033045 |
A037456 |
A037462 |
A037467 |
A037471 |
A037474 |
— |
|
A028903
|
| q=9 |
A033046 |
|
A037463 |
A037468 |
A037472 |
A037475 |
A037477 |
— |
A028904
|
| q=10 |
A007088 |
A007089 |
A007090 |
A007091 |
A007092 |
A007093 |
A007094 |
A007095
A0374792 |
—
|
| q=11 |
A033047 |
|
|
|
|
|
|
|
|
| q=12 |
A033048 |
|
|
|
|
|
|
|
A1024871
|
| q=13 |
A033049 |
|
|
|
|
|
|
|
A094823
|
| q=14 |
A033050 |
|
|
|
|
|
|
|
|
| q=15 |
A033051 |
|
|
|
|
|
|
|
|
| q=16 |
A033052 |
|
|
|
|
|
|
|
A102489
|
| q=17 |
A197351 |
|
|
|
|
|
|
|
|
| q=18 |
A197352 |
|
|
|
|
|
|
|
|
| q=19 |
A197353 |
|
|
|
|
|
|
|
|
| q=20 |
A063012 |
|
|
|
|
|
|
|
A1024911
|
1 These sequences have offset 1 and start with n=0.
2 These sequences have offset 1 and start with n=1.
All other sequences have offset 0 and start with n=0.
Sums of distinct powers of q
The first column (b=2) of the table above shows the sequences for Sums of distinct powers of q, since the binary digits in n enumerate all such powers.
Examples
A037454: 3[n]6
n = 0 1 2 3 4 5 6 7 8 9 10 11
a(n) = 0, 1, 2, 6, 7, 8, 12, 13, 14, 36, 37, 38, ...
n = 11: 1110 = 1023 -> 1026 = 1*6^2 + 0*6^1 + 2*6^0 = 3810 = a(11)
Programs
b:=3; q:=6; Table[FromDigits[RealDigits[n, b], q], {n, 0, 100}]
b=3; q=6; for(n=0,100,print1(fromdigits(digits(n, b), q),","));
java -cp joeis.jar irvine.oeis.a037.A037454 3 6