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A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

%I A383706 #9 May 18 2025 09:58:29
%S A383706 1,1,1,0,2,1,2,0,0,1,3,0,4,1,1,0,5,0,6,0,2,2,8,0,2,2,0,0,10,1,12,0,2,
%T A383706 3,2,0,15,3,2,0,18,1,22,0,0,5,27,0,2,0,3,0,32,0,3,0,4,5,38,0,46,7,0,0,
%U A383706 4,1,54,0,5,1,64,0,76,8,0,0,3,1,89,0,0,10
%N A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.
%C A383706 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.
%e A383706 The prime indices of 25 are (3,3), for which we have choices ((3),(2,1)) and ((2,1),(3)), so a(25) = 2.
%e A383706 The prime indices of 91 are (4,6), for which we have choices ((4),(6)), ((4),(5,1)), ((4),(3,2,1)), ((3,1),(6)), ((3,1),(4,2)), so a(91) = 5.
%e A383706 The prime indices of 273 are (2,4,6), for which we have choices ((2),(4),(6)), ((2),(4),(5,1)), ((2),(3,1),(6)), so a(273) = 3.
%t A383706 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
%t A383706 prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383706 Table[Length[pof[prix[n]]],{n,100}]
%Y A383706 Adding up over all integer partitions gives A279790, strict A279375.
%Y A383706 Without disjointness we have A357982, non-strict version A299200.
%Y A383706 For multiplicities instead of indices we have A382525.
%Y A383706 Positions of 0 appear to be A382912, counted by A383710, odd case A383711.
%Y A383706 Positions of positive terms are A382913, counted by A383708, odd case A383533.
%Y A383706 Positions of 1 are A383707, counted by A179009.
%Y A383706 The conjugate version is A384005.
%Y A383706 A000041 counts integer partitions, strict A000009.
%Y A383706 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A383706 A055396 gives least prime index, greatest A061395.
%Y A383706 A056239 adds up prime indices, row sums of A112798.
%Y A383706 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A383706 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A383706 Cf. A098859, A122111, A130091, A209229, A317141, A381454, A382771, A382876.
%K A383706 nonn
%O A383706 1,5
%A A383706 _Gus Wiseman_, May 15 2025