A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 3, 0, 1, 1, 4, 3, 0, 2, 2, 2, 0, 3, 4, 3, 1, 2, 6, 3, 1, 0, 5, 4, 2, 2, 4, 3, 0, 2, 3, 5, 0, 1, 3, 4, 3, 2, 4, 3, 3, 4, 5, 4, 0, 2, 5, 5, 0, 4, 6, 2, 1, 1, 7, 3, 1, 2, 7, 4, 2, 4, 5, 5, 1, 3, 6, 5, 1, 3, 6, 6, 3, 4, 4, 4, 2, 4, 7, 6, 3, 1, 4, 4, 0, 4, 6, 5, 2, 2, 7, 5, 2, 1, 7, 8, 4

Offset

0,11

Comments

Conjecture: a(n) > 0 for all n > 108.

Links

Table of n, a(n) for n=0..120.

Ilya Gutkovskiy, Extended graphical example

Eric Weisstein's World of Mathematics, Prime Power

Eric Weisstein's World of Mathematics, Squarefree

Formula

G.f.: (Sum_{k>=1} x^A246547(k))*(Sum_{k>=1} x^A000469(k)).

Example

a(26) = 3 because we have [25, 1], [22, 4] and [16, 10].

Mathematica

nmax = 120; CoefficientList[Series[(Sum[Boole[SquareFreeQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]) (Sum[Boole[PrimePowerQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Prog

(PARI) x='x+O('x^120); concat([0, 0, 0, 0, 0], Vec(sum(k=1, 120, (issquarefree(k) && !isprime(k))*x^k) * sum(k=1, 120, (isprimepower(k) && !isprime(k))*x^k))) \\ Indranil Ghosh, May 23 2017

Crossrefs

Cf. A000469, A098983, A246547, A282192, A282290, A282318, A282947.

Sequence in context: A285796 A317241 A361499 * A100699 A108921 A071548

Adjacent sequences: A287296 A287297 A287298 * A287300 A287301 A287302

Keyword

nonn

Author

Ilya Gutkovskiy, May 23 2017

Status

approved