cp's OEIS Frontend

A387326 Numbers whose prime factors do not have choosable sets of integer partitions.

%I A387326 #8 Sep 06 2025 11:07:18
%S A387326 8,16,24,32,40,48,56,64,72,80,81
%N A387326 Numbers whose prime factors do not have choosable sets of integer partitions.
%C A387326 We say that a set or 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.
%C A387326 Also numbers n with at least one prime index k such that the multiplicity of k in the prime factors of n exceeds the number of integer partitions of k.
%e A387326 The prime factors of 72 are {2,2,2,3,3}, with sets of partitions ({(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1,1),(2,1),(3)},{(1,1,1),(2,1),(3)}), which is not choosable, so 72 is in the sequence.
%t A387326 Select[Range[50],Select[Tuples[IntegerPartitions/@Join@@ConstantArray@@@FactorInteger[#]],UnsameQ@@#&]=={}&]
%Y A387326 The version for prime indices differs from A276079 in lacking 16807, counted by A387134.
%Y A387326 If we take the set {1..k} instead of the set of integer partitions of k we get A325127.
%Y A387326 A subset of A365886.
%Y A387326 Positions of zero in A387133.
%Y A387326 For prime indices instead of factors we have A387577.
%Y A387326 A000041 counts integer partitions, strict A000009.
%Y A387326 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387326 A387327 counts partitions of prime factors.
%Y A387326 A387328 counts partitions with choosable sets of partitions, ranks A387576.
%Y A387326 Cf. A063834, A120383, A289509, A299200, A335433, A335448, A355739, A383706, A387111, A387135, A387180.
%K A387326 nonn,more,new
%O A387326 1,1
%A A387326 _Gus Wiseman_, Sep 04 2025