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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 21 04:18 EST 2019. Contains 320371 sequences. (Running on oeis4.)