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A384394 Number of proper ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

%I A384394 #7 Jun 04 2025 10:25:13
%S A384394 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,
%T A384394 0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,
%U A384394 0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0
%N A384394 Number of proper ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.
%C A384394 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A384394 By "proper" we exclude the case of all singletons.
%e A384394 The prime indices of 216 are {1,1,1,2,2,2}, with conjugate partition (6,3), with proper choices ((6),(2,1)), ((5,1),(3)), and ((4,2),(3)), so a(216) = 3.
%t A384394 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384394 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A384394 pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
%t A384394 Table[Length[pofprop[conj[prix[n]]]],{n,100}]
%Y A384394 Conjugate prime indices are the rows of A122111.
%Y A384394 The non-proper version is A384005, conjugate A383706.
%Y A384394 This is the conjugate version of A384389 (firsts A384396).
%Y A384394 A000041 counts integer partitions, strict A000009.
%Y A384394 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384394 A055396 gives least prime index, greatest A061395.
%Y A384394 A056239 adds up prime indices, row sums of A112798.
%Y A384394 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384394 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384394 See also A382912, counted by A383710, odd case A383711.
%Y A384394 See also A382913, counted by A383708, odd case A383533.
%Y A384394 Cf. A279790, A357982, A381454, A382525, A384321, A384322, A384347, A384349, A384390, A384393.
%K A384394 nonn
%O A384394 1,27
%A A384394 _Gus Wiseman_, Jun 03 2025