A385576 Numbers whose prime indices have the same number of distinct elements as maximal anti-runs.

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 163

Offset: 1

Author

Gus Wiseman, Jul 04 2025

Keywords

• nonn

Comments

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.

These are also numbers with the same number of adjacent equal prime indices as adjacent unequal prime indices.

Examples

			The prime indices of 2640 are {1,1,1,1,2,3,5}, with 4 distinct parts {1,2,3,5} and 4 maximal anti-runs ((1),(1),(1),(2,3,5)), so 2640 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  28: {1,1,4}
  29: {10}
  31: {11}
  37: {12}
  41: {13}
  43: {14}
  44: {1,1,5}
  45: {2,2,3}
  47: {15}
		

Crossrefs

The LHS is the rank statistic A001221, triangle counted by A116608.

The RHS is the rank statistic A375136, triangle counted by A133121.

These partitions are counted by A385574.

A034296 counts flat or gapless partitions, ranks A066311 or A073491.

A047993 counts partitions with max part = length, ranks A106529.

A356235 counts partitions with a neighborless singleton, ranks A356237.

A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.

A384893 counts subsets by maximal anti-runs, for partitions A268193, strict A384905.

A385572 counts subsets with the same number of runs as anti-runs, ranks A385575.

Cf. A044813, A046660, A210034, A297155, A356226, A361205, A384889, A385213.

Programs

Mathematica

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

Formula

A001221(a(n)) = A375136(a(n)).