A361177 Exponentially powerful numbers: numbers whose exponents in their canonical prime factorization are all powerful numbers (A001694).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Author

Amiram Eldar, Mar 03 2023

Keywords

• nonn

Comments

First differs from it subsequence A197680 at n = 167: a(167) = 256 is not a term of A197680.

The asymptotic density of this sequence is Product_{p prime} ((1 - 1/p)*(1 + Sum_{i>=1} 1/p^A001694(i))) = 0.6427901996... .

Links

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

Crossrefs

Cf. A001694.

Similar sequences: A197680, A209061, A138302, A268335.

Programs

Mathematica

powQ[n_] := n == 1 || Min[FactorInteger[n][[;; , 2]]] > 1; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], powQ] &]

PARI

ispow(n) = {n == 1 || vecmin(factor(n)[,2]) > 1; }
is(n) = {my(e = factor(n)[, 2]); if(n == 1, return(1)); for(i=1, #e, if(!ispow(e[i]), return(0))); 1;}