A384011 Numbers k such that it is not possible to choose disjoint strict integer partitions of each conjugate prime index of k.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Author

Gus Wiseman, May 23 2025

Keywords

• nonn

Comments

A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.

Examples

			The terms together with their prime indices begin:
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   28: {1,1,4}
		

Crossrefs

The conjugate is A382912.

These complement is counted by A383708, ranks A382913 or A384010.

These partitions are counted by A383710, conjugate A383711.

These are the positions of 0 in A384005, conjugate A383706.

A000041 counts integer partitions, strict A000009.

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represent conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.

A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.

Cf. A098859, A130091, A279375, A279790, A357982, A382525, A383533, A383707.

Programs

Mathematica

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],pof[conj[prix[#]]]=={}&]