A286971 - OEIS

A286971

Number of ways to write n as a sum of two numbers, one of which is the product of an even number of distinct primes (including 1) (A030229) and another is the product of an odd number of distinct primes (A030059).

0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 2, 2, 1, 1, 1, 4, 2, 2, 2, 2, 1, 3, 3, 3, 2, 3, 3, 4, 1, 3, 3, 4, 2, 3, 3, 5, 5, 4, 5, 5, 3, 5, 6, 6, 4, 3, 4, 4, 3, 7, 7, 6, 3, 3, 6, 8, 6, 4, 4, 3, 8, 8, 8, 7, 2, 7, 10, 8, 5, 5, 6, 4, 8, 8, 12, 7, 3, 7, 11, 11, 8, 3, 7, 9, 6, 10, 14, 8, 4, 5, 12, 13, 10, 7, 9, 8, 12, 13, 12

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OFFSET

0,9

COMMENTS

Conjecture: a(n) > 0 for all n > 10.

LINKS

Table of n, a(n) for n=0..100.

FORMULA

G.f.: (Sum_{i>=1} x^A030229(i))*(Sum_{j>=1} x^A030059(j)).

EXAMPLE

a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].

MATHEMATICA

nmax = 100; CoefficientList[Series[(Sum[Boole[MoebiusMu[k] == 1] x^k, {k, 1, nmax}]) (Sum[Boole[MoebiusMu[k] == -1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

CROSSREFS

Cf. A005117, A030059, A030229, A098235, A098236, A285796, A285797.

Sequence in context: A076191 A362955 A282318 * A025861 A090723 A027357

Adjacent sequences: A286968 A286969 A286970 * A286972 A286973 A286974

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 17 2017

STATUS

approved