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

1,2

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

Links

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

Example

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}

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]]&]

Crossrefs

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

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

Complement of A276936.

Sequence in context: A137292 A089358 A173001 * A047500 A359822 A257408

Adjacent sequences: A325127 A325128 A325129 * A325131 A325132 A325133

Keyword

nonn

Author

Gus Wiseman, Apr 01 2019

Status

approved