A283762 Expansion of (x + Sum_{k>=1} x^prime(k))^3.

0, 0, 0, 1, 3, 6, 7, 9, 9, 13, 12, 15, 9, 15, 12, 22, 15, 24, 12, 27, 18, 34, 18, 36, 15, 42, 24, 45, 15, 42, 12, 51, 24, 52, 18, 60, 21, 66, 24, 58, 15, 69, 18, 75, 30, 75, 24, 87, 21, 93, 36, 91, 24, 99, 18, 108, 36, 97, 18, 108, 21, 126, 42, 111, 21, 135, 30, 141, 36, 112, 18, 150, 30, 153, 42, 138, 33, 177, 30, 171, 42

Offset: 0

Author

Ilya Gutkovskiy, Mar 16 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 3 noncomposite numbers (1 together with the primes) (A008578).

a(2k+1) > 0 for all k > 0 (from the ternary Goldbach's conjecture, proved by H. A. Helfgott).

Examples

			a(6) = 7 because we have [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 2, 2], [2, 1, 3], [1, 3, 2] and [1, 2, 3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Ilya Gutkovskiy, Extended graphical example

Crossrefs

Cf. A008578, A068307, A098238, A124867.

Programs

Mathematica

nmax = 80; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^3, {x, 0, nmax}], x]

PARI

concat([0, 0, 0], Vec((x + sum(k=1, 80, x^prime(k)))^3 + O(x^81))) \\ Indranil Ghosh, Mar 16 2017

Formula

G.f.: (x + Sum_{k>=1} x^prime(k))^3.