cp's OEIS Frontend

A387576 Numbers k such that it is possible to choose a sequence of distinct integer partitions, one of each prime index of k (with multiplicity).

%I A387576 #15 Sep 09 2025 20:57:23
%S A387576 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 A387576 35,37,38,39,41,42,43,45,46,47,49,50,51,53,55,57,58,59,61,62,63,65,66,
%U A387576 67,69,70,71,73,74,75,77,78,79,82,83,85,86,87,89,90,91,93
%N A387576 Numbers k such that it is possible to choose a sequence of distinct integer partitions, one of each prime index of k (with multiplicity).
%C A387576 First differs from A276078 in having 16807.
%C A387576 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 A387576 Also numbers with no prime factor prime(x) whose exponent exceeds A000041(x).
%e A387576 The prime indices of 125 are (3,3,3), and we have choices such as ((3),(1,1,1),(2,1)), so 125 is included.
%e A387576 The terms together with their prime indices begin:
%e A387576    1: {}
%e A387576    2: {1}
%e A387576    3: {2}
%e A387576    5: {3}
%e A387576    6: {1,2}
%e A387576    7: {4}
%e A387576    9: {2,2}
%e A387576   10: {1,3}
%e A387576   11: {5}
%e A387576   13: {6}
%e A387576   14: {1,4}
%e A387576   15: {2,3}
%e A387576   17: {7}
%e A387576   18: {1,2,2}
%t A387576 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A387576 Select[Range[100],Select[Tuples[IntegerPartitions/@prix[#]],UnsameQ@@#&]!={}&]
%Y A387576 For initial intervals instead of partitions we have A276078, complement A276079.
%Y A387576 For divisors instead of partitions we have A368110, counted by A239312.
%Y A387576 These are positions of nonzero terms in A387115.
%Y A387576 For sets of strict partitions we have A387177, complement A387176.
%Y A387576 Integer partitions of this type are counted by A387328.
%Y A387576 The complement is A387577, counted by A387134.
%Y A387576 A000041 counts integer partitions, strict A000009.
%Y A387576 A003963 multiplies together the prime indices of n.
%Y A387576 A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
%Y A387576 A238873 counts superdiagonal partitions, ranks A387112.
%Y A387576 A239312 counts partitions with choosable divisors, strict A387178.
%Y A387576 A289509 lists numbers with relatively prime prime indices.
%Y A387576 A370320 counts partitions without choosable divisors, strict A387137.
%Y A387576 A387118 counts non-superdiagonal partitions, ranks A387113.
%Y A387576 Cf. A000720, A052335, A052337, A120383, A335433, A335448, A355740, A367771, A383706, A387110, A387181.
%K A387576 nonn,new
%O A387576 1,2
%A A387576 _Gus Wiseman_, Sep 07 2025