A257497 - OEIS

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A257497

Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.

1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 5, 2, 3, 4, 4, 4, 2, 2, 3, 4, 6, 3, 2, 5, 7, 5, 2, 4, 3, 5, 4, 3, 4, 4, 6, 5, 3, 3, 5, 4, 5, 2, 2, 5, 4, 4, 2, 3, 5, 5, 6, 1, 4, 5, 4, 3, 3, 7, 4, 2, 5, 2, 5, 4, 2, 4, 3, 6, 4, 5, 9, 4, 3, 3, 4, 8, 2, 4, 5, 3, 5, 1, 5, 4, 1, 5, 3, 2, 4, 6, 6, 3, 5, 4, 6, 5, 5, 5

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OFFSET

1,2

COMMENTS

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 54, 84, 87, 109, 174, 252, 344, 1234, 1439, 2924.

This implies the Twin Prime Conjecture.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.

EXAMPLE

a(1439) = 1 since 1439 = 1424 + 15 = floor(4274/3) + (-3)*(3*(-3)-1)/2 with {3*4274-1,3*4274+1} = {12821,12823} a twin prime pair.

a(2924) = 1 since 2924 = 2334 + 590 = floor(7004/3) + 20*(3*20-1)/2 with {3*7004-1, 3*7004+1} = {21011,21013} a twin prime pair.