A325130 Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.

1, 3, 4, 5, 7, 8, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

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 the integer partitions counted by A276429.

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.68974964705635552968... - Amiram Eldar, Jan 09 2021

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
  11: {5}
  12: {1,1,2}
  13: {6}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A056239, A087153, A112798, A124010, A276078, A276429.

Cf. A324525, A324571, A325127, A325128, A325130, A325131.

Complement of A276936.

Programs

Maple

q:= n-> andmap(i-> numtheory[pi](i[1])<>i[2], ifactors(n)[2]):
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while not q(k) do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 28 2019

Mathematica

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