A280130 Expansion of Product_{k>=2} (1 + x^(k^3)).
1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0
Keywords
Examples
a(35) = 1 because 35 = 27 + 8. This is the first nonzero value for a noncube index. From _Antti Karttunen_, Aug 30 2017: (Start) a(72) = 1 because there is just one solution: 72 = 4^3 + 2^3. a(216) = 2 because there are two solutions: 216 = 6^3 = 5^3 + 4^3 + 3^3. This is the first index where a(n) > 1. (End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..100000 (first 10001 terms from Antti Karttunen)
- Index entries for sequences related to sums of cubes
- Index entries for related partition-counting sequences
Programs
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Mathematica
nmax = 130; CoefficientList[Series[Product[1 + x^k^3, {k, 2, nmax}], {x, 0, nmax}], x] -
PARI
A280130(n,m=2) = { my(s=0); if(!n,1,for(c=m,n,if(ispower(c,3), s+=A280130(n-c,c+1))); (s)); }; \\ Antti Karttunen, Aug 30 2017
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PARI
A280130(n,m=2)={if(n, sum(c=m, sqrtnint(n,3), A280130(n-c^3, c+1)), 1)} \\ At n ~ 2500 this is about 100 times faster than code from 2017, but for larger n (needed for A030272(n)=a(n^3)) better use (with parisize (or allocmem) >= 201*Nmax): V280130=Vecsmall(prod(k=2, (Nmax=3*10^4)^(1/3), 1+x^k^3+O(x^Nmax))); A280130(n)=V280130[n+1] \\ M. F. Hasler, Jan 05 2020
Formula
G.f.: Product_{k>=2} (1 + x^(k^3)).
From Vaclav Kotesovec, Dec 26 2016: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * A279329(k).
a(n) + a(n-1) = A279329(n).
a(n) ~ A279329(n)/2.
(End)
Extensions
Secondary offset added by Antti Karttunen, Jul 07 2025
Comments