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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

%I

%S 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,

%T 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,

%U 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

%N 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).

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

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

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

%t 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]

%Y Сf. A005117, A030059, A030229, A098235, A098236, A285796, A285797.

%K nonn

%O 0,9

%A _Ilya Gutkovskiy_, May 17 2017

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Last modified October 23 20:44 EDT 2018. Contains 316530 sequences. (Running on oeis4.)