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A387181 Heinz numbers of integer partitions with no part k appearing more than A000005(k) times.

%I A387181 #9 Sep 09 2025 08:11:21
%S A387181 1,2,3,5,6,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26,29,30,31,33,34,
%T A387181 35,37,38,39,41,42,43,45,46,47,49,50,51,53,55,57,58,59,61,62,63,65,66,
%U A387181 67,69,70,71,73,74,75,77,78,79,82,83,85,86,87,89,90,91,93,94,95,97,98,99,101,102,103,105,106,107,109,110,111,113,114,115,117,118,119,121,122,123,126
%N A387181 Heinz numbers of integer partitions with no part k appearing more than A000005(k) times.
%C A387181 First differs from A276078 in lacking 125.
%C A387181 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.
%C A387181 Also numbers of which it is possible to choose a different constant integer partition of each prime index. The multiset of prime indices of n is row n of A112798.
%e A387181 The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
%e A387181   ((1),(1),(2),(3))
%e A387181   ((1),(1),(2),(1,1,1))
%e A387181   ((1),(1),(1,1),(3))
%e A387181   ((1),(1),(1,1),(1,1,1))
%e A387181 Since none of these is strict, 60 is not in the sequence.
%e A387181 The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
%e A387181   ((1),(2),(1,1),(3))
%e A387181   ((1),(2),(1,1),(1,1,1))
%e A387181   ((1),(1,1),(2),(3))
%e A387181   ((1),(1,1),(2),(1,1,1))
%e A387181 So 90 is in the sequence.
%t A387181 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387181 Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[#]],UnsameQ@@#&]!={}&]
%t A387181 - or -
%t A387181 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387181 Select[Range[100],And@@Table[Count[prix[#],k]<=DivisorSigma[0,k],{k,Union[prix[#]]}]&]
%Y A387181 The complement for divisors is A355740, counted by A370320.
%Y A387181 The complement for prime factors is A355529, counted by A370593.
%Y A387181 For prime factors instead of constant partitions we have A368100, counted by A370592.
%Y A387181 For divisors instead of constant partitions we have A368110, counted by A239312.
%Y A387181 These are all positions of nonzero terms in A387120.
%Y A387181 The complement for strict partitions is A387176, counted by A387137.
%Y A387181 For strict instead of constant partitions we have A387177, counted by A387178.
%Y A387181 Twice-partitions of this type are counted by A387179, constant-block case of A296122.
%Y A387181 The complement is A387180, counted by A387329.
%Y A387181 Partitions of this type are counted by A387330.
%Y A387181 A000041 counts integer partitions, strict A000009.
%Y A387181 A003963 multiplies together prime indices.
%Y A387181 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387181 A120383 lists numbers divisible by all of their prime indices.
%Y A387181 A289509 lists numbers with relatively prime prime indices.
%Y A387181 Cf. A000005, A052335, A063834, A238873, A276079, A335433, A335448, A355739, A383706, A387110.
%K A387181 nonn,new
%O A387181 1,2
%A A387181 _Gus Wiseman_, Sep 04 2025