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A340756 Number of partitions of n into 4 semiprime parts.

%I A340756 #11 May 14 2022 13:45:32
%S A340756 1,0,1,0,1,1,2,1,2,1,3,3,4,2,3,4,5,6,5,4,7,7,9,9,9,7,9,12,13,11,11,13,
%T A340756 16,17,17,18,18,17,20,25,25,23,24,26,32,29,31,33,31,33,35,43,43,40,39,
%U A340756 45,48,52,50,52,53,52,61,69,67,61,61,70,79,76,76,80,81,85,88,101
%N A340756 Number of partitions of n into 4 semiprime parts.
%H A340756 Alois P. Heinz, <a href="/A340756/b340756.txt">Table of n, a(n) for n = 16..10000</a>
%H A340756 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A340756 a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} [Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.
%F A340756 a(n) = [x^n y^4] 1/Product_{j>=1} (1-y*x^A001358(j)). - _Alois P. Heinz_, May 21 2021
%t A340756 Table[Sum[Sum[Sum[KroneckerDelta[PrimeOmega[k], PrimeOmega[j], PrimeOmega[i], PrimeOmega[n - i - j - k], 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 16, 100}]
%t A340756 Table[Count[IntegerPartitions[n,{4}],_?(PrimeOmega[#]=={2,2,2,2}&)],{n,16,95}] (* _Harvey P. Dale_, May 14 2022 *)
%Y A340756 Cf. A001222 (Omega), A001358.
%Y A340756 Column k=4 of A344447.
%K A340756 nonn
%O A340756 16,7
%A A340756 _Wesley Ivan Hurt_, Jan 19 2021