OFFSET
1,15
COMMENTS
Conjecture: For every k = 2, 3, ... there is a positive integer N(k) such that any integer n > N(k) can be written as n_1 + n_2 + ... + n_k with n_1, n_2, ..., n_k positive and distinct such that the product sigma(n_1)*sigma(n_2)*...*sigma(n_k) is a k-th power. In particular, we may take N(2) = 309, N(3) = 42, N(4) = 25, N(5) = 24, N(6) = 27 and N(7) = 32.
This is similar to the conjecture in A233386.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
EXAMPLE
a(9) = 1 since 9 = 1 + 1 + 7 with sigma(1)*sigma(1)*sigma(7) = 1*1*8 = 2^3.
a(41) = 1 since 41 = 2 + 6 + 33 with sigma(2)*sigma(6)*sigma(33) = 3*12*48 = 12^3.
a(50) = 1 since 50 = 2 + 17 + 31 with sigma(2)*sigma(17)*sigma(31) = 3*18*32 = 12^3.
MATHEMATICA
sigma[n_]:=DivisorSigma[1, n]
CQ[n_]:=IntegerQ[n^(1/3)]
p[i_, j_, k_]:=CQ[sigma[i]*sigma[j]*sigma[k]]
a[n_]:=Sum[If[p[i, j, n-i-j], 1, 0], {i, 1, (n-1)/3}, {j, i, (n-i)/2}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 02 2014
STATUS
approved