A237050 Number of ways to write n = i_1 + i_2 + i_3 + i_4 + i_5 (0 < i_1 <= i_2 <= i_3 <= i_4 <= i_5) with i_1, i_2, ..., i_5 not all equal such that the product i_1*i_2*i_3*i_4*i_5 is a fifth power.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 4, 3, 3, 3, 5, 4, 5, 7, 3, 5, 3, 4, 3, 3, 4, 6, 4, 4, 4, 4, 2, 4, 3, 5, 5, 3, 5, 4, 8, 7, 7, 9, 10, 9, 12, 7, 6, 9, 10, 9, 9, 8, 8, 7, 10, 7, 10, 10, 10, 10, 5, 8, 13, 10, 9, 8, 12, 15, 10, 12, 9, 8

Offset

1,31

Comments

Conjecture: For each k = 3, 4, ... there is a positive integer M(k) such that any integer n > M(k) can be written as i_1 + i_2 + ... + i_k with i_1, i_2, ..., i_k positive and not all equal such that the product i_1*i_2*...*i_k is a k-th power. In particular, we may take M(3) = 486, M(4) = 23, M(5) = 26, M(6) = 36 and M(7) = 31.

This is motivated by the conjectures in A233386 and A237049.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Tianxin Cai and Deyi Chen, A new variant of the Hilbert-Waring problem, Math. Comp. 82 (2013), 2333-2341.

Example

a(25) = 1 since 25 = 1 + 4 + 4 + 8 + 8 with 1*4*4*8*8 = 4^5.

Mathematica

QQ[n_]:=IntegerQ[n^(1/5)]
a[n_]:=Sum[If[QQ[i*j*h*k*(n-i-j-h-k)], 1, 0], {i, 1, (n-1)/5}, {j, i, (n-i)/4}, {h, j, (n-i-j)/3}, {k, h, (n-i-j-h)/2}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000584, A233386, A236998, A237016, A237049.

Sequence in context: A216200 A157873 A022870 * A235130 A131410 A202453

Adjacent sequences: A237047 A237048 A237049 * A237051 A237052 A237053

Keyword

nonn

Author

Zhi-Wei Sun, Feb 02 2014

Status

approved