cp's OEIS Frontend

A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

%I A387177 #7 Aug 31 2025 10:46:53
%S A387177 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,25,26,29,30,31,33,34,35,37,
%T A387177 38,39,41,42,43,46,47,49,50,51,53,55,57,58,59,61,62,65,66,67,69,70,71,
%U A387177 73,74,75,77,78,79,82,83,85,86,87,89,91,93,94,95,97,98
%N A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.
%C A387177 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 A387177 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.
%e A387177 The prime indices of 50 are {1,3,3}, and {(1),(3),(2,1)} is a valid choice of distinct strict partitions, so 50 is in the sequence.
%t A387177 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387177 strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
%t A387177 Select[Range[100],Select[Tuples[strptns/@prix[#]],UnsameQ@@#&]!={}&]
%Y A387177 The version for all partitions appears to be A276078, counted by A052335.
%Y A387177 The complement for all partitions appears to be A276079, counted by A387134.
%Y A387177 The complement for divisors is A355740, counted by A370320.
%Y A387177 Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
%Y A387177 The version for divisors is A368110, counted by A239312.
%Y A387177 The version for initial intervals is A387112, counted by A238873, see A387111.
%Y A387177 The complement for initial intervals is A387113, counted by A387118.
%Y A387177 These are the positions of nonzero terms in A387115.
%Y A387177 The complement is A387176.
%Y A387177 Partitions of this type are counted by A387178, complement A387137.
%Y A387177 The complement for constant partitions is A387180, counted by A387329, see A387120.
%Y A387177 The version for constant partitions is A387181, counted by A387330.
%Y A387177 A000041 counts integer partitions, strict A000009.
%Y A387177 A003963 multiplies together the prime indices of n.
%Y A387177 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387177 A289509 lists numbers with relatively prime prime indices.
%Y A387177 Cf. A000720, A120383, A270995, A299200, A335433, A335448, A357978, A357982, A383706, A387110.
%K A387177 nonn,new
%O A387177 1,2
%A A387177 _Gus Wiseman_, Aug 29 2025