A325128 Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141

Offset: 1

Author

Gus Wiseman, Apr 01 2019

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears fewer than k times. Such partitions are counted by A087153.

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k) = 0.44070243286030291209... - Amiram Eldar, Feb 02 2021

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
  47: {15}
  49: {4,4}
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A056239, A062457, A087153, A109298, A112798, A115584, A118914, A276078.

Cf. A324524, A324525, A324571, A325127, A325130.

Programs

Mathematica

Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k