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Heinz numbers of integer partitions of prime numbers into prime parts.
5

%I #5 Jun 25 2018 22:51:25

%S 3,5,11,15,17,31,33,41,45,59,67,83,93,109,127,153,157,177,179,191,211,

%T 241,275,277,283,297,327,331,353,367,369,375,401,405,425,431,459,461,

%U 509,537,547,563,587,599,603,605,617,709,739,773,775,797,825,831,837

%N Heinz numbers of integer partitions of prime numbers into prime parts.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%e Sequence of integer partitions of prime numbers into prime parts together with their Heinz numbers begins:

%e 3: (2)

%e 5: (3)

%e 11: (5)

%e 15: (2,3)

%e 17: (7)

%e 31: (11)

%e 33: (2,5)

%e 41: (13)

%e 45: (2,2,3)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],And[PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&

%Y Cf. A000041, A000607, A056239, A056768, A076610, A100118, A112798, A215366, A296150, A300383, A316091.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 24 2018