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A316092 Heinz numbers of integer partitions of prime numbers into prime parts.

%I A316092 #9 Jun 19 2025 09:19:04
%S A316092 3,5,11,15,17,31,33,41,45,59,67,83,93,109,127,153,157,177,179,191,211,
%T A316092 241,275,277,283,297,327,331,353,367,369,375,401,405,425,431,459,461,
%U A316092 509,537,547,563,587,599,603,605,617,709,739,773,775,797,825,831,837
%N A316092 Heinz numbers of integer partitions of prime numbers into prime parts.
%C A316092 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A316092 Sequence of integer partitions of prime numbers into prime parts together with their Heinz numbers begins:
%e A316092    3: (2)
%e A316092    5: (3)
%e A316092   11: (5)
%e A316092   15: (2,3)
%e A316092   17: (7)
%e A316092   31: (11)
%e A316092   33: (2,5)
%e A316092   41: (13)
%e A316092   45: (2,2,3)
%t A316092 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A316092 Select[Range[900],And[PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]
%Y A316092 Cf. A000041, A000607, A056239, A056768, A076610, A100118, A112798, A215366, A296150, A300383, A316091.
%K A316092 nonn
%O A316092 1,1
%A A316092 _Gus Wiseman_, Jun 24 2018