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%I #7 Oct 17 2022 12:32:10
%S 1,1,1,3,1,3,1,6,2,3,1,7,1,3,3,11,1,7,1,8,3,3,1,14,3,3,4,8,1,11,1,19,
%T 3,3,3,18,1,3,3,18,1,12,1,8,8,3,1,27,3,10,3,8,1,16,3,19,3,3,1,25,1,3,
%U 8,33,3,12,1,8,3,12,1,35,1,3,11,8,3,12,1,34,9
%N Number of integer partitions that can be obtained by iteratively adding and multiplying together parts of the integer partition with Heinz number n.
%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%e The a(n) partitions for n = 1, 4, 8, 9, 12, 16, 20, 24:
%e () (1) (1) (4) (2) (1) (3) (2)
%e (2) (2) (22) (3) (2) (4) (3)
%e (11) (3) (4) (3) (5) (4)
%e (11) (21) (4) (6) (5)
%e (21) (22) (11) (31) (6)
%e (111) (31) (21) (32) (21)
%e (211) (22) (41) (22)
%e (31) (311) (31)
%e (111) (32)
%e (211) (41)
%e (1111) (211)
%e (221)
%e (311)
%e (2111)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t Table[Length[ReplaceListRepeated[{primeMS[n]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}]],{n,100}]
%Y The single-part partitions are counted by A319841, with an inverse A319913.
%Y The minimum is A319855, maximum A319856.
%Y A000041 counts integer partitions.
%Y A001222 counts prime indices, distinct A001221.
%Y A056239 adds up prime indices.
%Y A066739 counts representations as a sum of products.
%Y Cf. A000792, A001055, A001970, A005520, A048249, A063834, A066815, A318948, A319850, A319909, A319910.
%K nonn
%O 1,4
%A _Gus Wiseman_, Oct 17 2022