A025447 -id:A025447 - cp's OEIS frontend

A025453 Number of partitions of n into 9 nonnegative cubes.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 4, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 4, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3

Offset: 0

David W. Wilson

nonn

			a(8) = 2 via 8*0^3 + 1*2^3 = 1 * 0^3 + 8*1^3.

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A025446, A025447, A025448, A025449, A025450, A025451, A025452, this sequence, A025454.

f:= proc(x,m,M)
local i;
option remember;
  if x = 0 then return 1
  elif m = 0 then return 0
  fi;
  add(procname(x-i^3, m-1, i), i=1..min(M,floor(x^(1/3))));
end proc:
map(f, [$0..150],9,150); # Robert Israel, Jan 23 2025

first(n) = my(v=vector(n), maxb=sqrtnint(n, 3)); forvec(x=vector(9, i, [0, maxb]), s=sum(i=1, 9, x[i]^3); if(0David A. Corneth, Jan 23 2025

A001239 Numbers that are the sum of 3 nonnegative cubes in more than 1 way.

Original entry on oeis.org

216, 251, 344, 729, 855, 1009, 1072, 1366, 1457, 1459, 1520, 1674, 1728, 1729, 1730, 1737, 1756, 1763, 1793, 1854, 1945, 2008, 2072, 2241, 2414, 2456, 2458, 2729, 2736, 2752, 3060, 3391, 3402, 3457, 3500, 3592, 3599, 3655, 3744, 3745

Offset: 1

N. J. A. Sloane

nonn

G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 165.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Decomposition of first 10000 terms into cube triples

Cf. A001235, A003998, A008917, A025396, A025456, A025447.

Select[Range[4000], Length[PowersRepresentations[#, 3, 3]] > 1 &] (* Harvey P. Dale, Feb 03 2011 *)

is(n)=my(t); for(a=0, sqrtnint(n, 3), my(a3=a^3, c); for(b=0, min(a, sqrtnint(n-a3, 3)), if(ispower(n-a3-b^3, 3, &c) && c <= b && t++>1, return(1)))); 0 \\ Charles R Greathouse IV, Jul 02 2017

A337366 Number of representations of A036691(n) as a sum of 3 nonnegative cubes.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 1, 2, 1, 4, 6, 3, 8, 8, 14, 7

Offset: 0

Altug Alkan, Aug 25 2020

Conjecture I: a(n) = 0 only for n = 1. That is, any product of first n > 1 composite numbers is a sum of at most 3 positive cubes. For example,
A036691(100) = 2563573191821442299652988946477367093137353211904000000000^3 + 21431289850849406740917647451954098598503667204096000000000^3 + 26409890400237152457638095665189553529771293409280000000000^3.
Conjecture II: For any term t >= 1, there are only finitely many values of n such that a(n) = t.

			a(4) = 2 because A036691(4) = 1728 = 12^3 = 6^3 + 8^3 + 10^3.

Cf. A025447, A036691, A226955, A267414.

A036691 = Join[{1}, FoldList[Times, Select[Range[20], CompositeQ]]];
Table[Length@ PowersRepresentations[A036691[[n]], 3, 3], {n, 10}] (* Robert Price, Sep 08 2020 *)

a(n) = A025447(A036691(n)).

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A025453 Number of partitions of n into 9 nonnegative cubes.

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A001239 Numbers that are the sum of 3 nonnegative cubes in more than 1 way.

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A337366 Number of representations of A036691(n) as a sum of 3 nonnegative cubes.

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