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A101048 Number of partitions of n into semiprimes (a(0) = 1 by convention). 12

%I

%S 1,0,0,0,1,0,1,0,1,1,2,0,2,1,3,2,3,1,5,3,5,4,7,4,9,7,10,8,13,10,17,13,

%T 18,17,25,21,29,25,34,34,43,37,51,49,61,59,73,69,89,87,103,103,124,

%U 122,148,149,172,176,206,208,244,248,281,293,337,344,391,405,456,479,537,553

%N Number of partitions of n into semiprimes (a(0) = 1 by convention).

%C Semiprime analog of A000607. a(n) <= A002095(n). - _Jonathan Vos Post_, Oct 01 2007

%H T. D. Noe, <a href="/A101048/b101048.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/product(product(1-x^(p(i)p(j)), i = 1..j),j = 1..infinity), p(k) is the k-th prime. - _Emeric Deutsch_, Apr 04 2006

%e a(12) = #{6 + 6, 4 + 4 + 4} = #{2 * (2*3), 3 * (2*2)} = 2.

%p g:=1/product(product(1-x^(ithprime(i)*ithprime(j)),i=1..j),j=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=1..71); # _Emeric Deutsch_, Apr 04 2006

%t terms = 100; CoefficientList[1/Product[1 - x^(Prime[i] Prime[j]), {i, 1, PrimePi[Ceiling[terms/2]]}, {j, 1, i}] + O[x]^terms, x] (* _Jean-Fran├žois Alcover_, Aug 01 2018 *)

%o (Haskell)

%o a101048 = p a001358_list where

%o p _ 0 = 1

%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Mar 21 2014

%Y Cf. A000041, A000607, A101049, A001358, A064911, A002095.

%Y Cf. A112020, A112021.

%Y Cf. A002100.

%K nonn

%O 0,11

%A _Reinhard Zumkeller_, Nov 28 2004

%E a(0) set to 1 by _N. J. A. Sloane_, Nov 23 2007

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Last modified February 23 07:28 EST 2020. Contains 332159 sequences. (Running on oeis4.)