cp's OEIS Frontend

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

%I A387176 #14 Aug 31 2025 10:47:56
%S A387176 4,8,9,12,16,18,20,24,27,28,32,36,40,44,45,48,52,54,56,60,63,64,68,72,
%T A387176 76,80,81,84,88,90,92,96,99,100,104,108,112,116,117,120,124,125,126,
%U A387176 128,132,135,136,140,144,148,152,153,156,160,162,164,168,171,172
%N A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.
%C A387176 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 A387176 We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
%t A387176 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387176 Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]
%Y A387176 The complement for all partitions appears to be A276078, counted by A052335.
%Y A387176 For all partitions we appear to have A276079, counted by A387134.
%Y A387176 For divisors instead of strict partitions we have A355740, counted by A370320.
%Y A387176 Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
%Y A387176 The complement for divisors is A368110, counted by A239312.
%Y A387176 The complement for initial intervals is A387112, counted by A238873, see A387111.
%Y A387176 For initial intervals instead of strict partitions we have A387113, counted by A387118.
%Y A387176 These are the positions of 0 in A387115.
%Y A387176 Partitions of this type are counted by A387137, complement A387178.
%Y A387176 The complement is A387177.
%Y A387176 The version for constant partitions is A387180, counted by A387329.
%Y A387176 The complement for constant partitions is A387181, counted by A387330.
%Y A387176 A000041 counts integer partitions, strict A000009.
%Y A387176 A003963 multiplies together the prime indices of n.
%Y A387176 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387176 A120383 lists numbers divisible by all of their prime indices.
%Y A387176 A289509 lists numbers with relatively prime prime indices.
%Y A387176 Cf. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.
%K A387176 nonn,new
%O A387176 1,1
%A A387176 _Gus Wiseman_, Aug 27 2025