A382856 Numbers whose prime indices do not have a mode of 1.

1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 108, 109, 111, 113, 115

Offset: 1

Author

Gus Wiseman, Apr 07 2025

Keywords

• nonn

Examples

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
		

Crossrefs

The case of non-unique mode is A024556.

The complement is A360015 except first.

Partitions of this type are are counted by A382526 except first, complement A241131.

A091602 counts partitions by the greatest multiplicity, rank statistic A051903.

A112798 lists prime indices, length A001222, sum A056239.

A116598 counts ones in partitions, rank statistic A007814.

A240312 counts partitions with max part = max multiplicity, ranks A381542.

A362611 counts modes in prime indices, triangle A362614.

For co-mode see A359178, A362613, A364061 (A364062), A364158 (A364159).

Cf. A000265, A002865, A106529, A327473, A327476, A362605, A363486, A356862, A360013, A360014, A381437.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],FreeQ[Commonest[prix[#]],1]&]