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Expansion of 1/(1 - Sum_{k>=2} floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).
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%I #8 Jan 09 2017 07:23:19

%S 1,0,0,0,1,0,1,0,1,1,3,0,2,2,6,3,6,3,11,10,16,10,23,23,40,34,52,52,93,

%T 94,130,133,209,234,330,352,488,570,804,909,1198,1405,1918,2283,2980,

%U 3512,4622,5636,7340,8811,11321,13864,17937,21957,27936,34262,43857,54290,68915,84940,107685,133811,169615,210375,265305

%N Expansion of 1/(1 - Sum_{k>=2} floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

%C Number of compositions (ordered partitions) into semiprimes (A001358).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: 1/(1 - Sum_{k>=2} floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k).

%e a(10) = 3 because we have [4, 6], [6, 4] and [10].

%t nmax = 44; CoefficientList[Series[1/(1 - Sum[Floor[PrimeOmega[k]/2] Floor[2/PrimeOmega[k]] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

%Y Cf. A001222, A001358, A023360, A101048.

%K nonn

%O 0,11

%A _Ilya Gutkovskiy_, Dec 29 2016