cp's OEIS Frontend

A357858 Number of integer partitions that can be obtained by iteratively adding and multiplying together parts of the integer partition with Heinz number n.

%I A357858 #7 Oct 17 2022 12:32:10
%S A357858 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 A357858 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 A357858 8,33,3,12,1,8,3,12,1,35,1,3,11,8,3,12,1,34,9
%N A357858 Number of integer partitions that can be obtained by iteratively adding and multiplying together parts of the integer partition with Heinz number n.
%C A357858 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 A357858 The a(n) partitions for n = 1, 4, 8, 9, 12, 16, 20, 24:
%e A357858   ()  (1)   (1)    (4)   (2)    (1)     (3)    (2)
%e A357858       (2)   (2)    (22)  (3)    (2)     (4)    (3)
%e A357858       (11)  (3)          (4)    (3)     (5)    (4)
%e A357858             (11)         (21)   (4)     (6)    (5)
%e A357858             (21)         (22)   (11)    (31)   (6)
%e A357858             (111)        (31)   (21)    (32)   (21)
%e A357858                          (211)  (22)    (41)   (22)
%e A357858                                 (31)    (311)  (31)
%e A357858                                 (111)          (32)
%e A357858                                 (211)          (41)
%e A357858                                 (1111)         (211)
%e A357858                                                (221)
%e A357858                                                (311)
%e A357858                                                (2111)
%t A357858 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357858 ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t A357858 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 A357858 The single-part partitions are counted by A319841, with an inverse A319913.
%Y A357858 The minimum is A319855, maximum A319856.
%Y A357858 A000041 counts integer partitions.
%Y A357858 A001222 counts prime indices, distinct A001221.
%Y A357858 A056239 adds up prime indices.
%Y A357858 A066739 counts representations as a sum of products.
%Y A357858 Cf. A000792, A001055, A001970, A005520, A048249, A063834, A066815, A318948, A319850, A319909, A319910.
%K A357858 nonn
%O A357858 1,4
%A A357858 _Gus Wiseman_, Oct 17 2022