A346068 Numbers that are the product of distinct primes with prime subscripts raised to prime powers.

1, 9, 25, 27, 121, 125, 225, 243, 289, 675, 961, 1089, 1125, 1331, 1681, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 6075, 6889, 7225, 7803, 8649, 11881, 11979, 15125, 15129, 16129, 24025, 24649, 25947, 27225, 28125, 29403, 29791, 30375, 31329, 32041, 33275, 34969

Offset: 1

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

• nonn

Examples

			675 = 3^3 * 5^2 = prime(prime(1))^prime(2) * prime(prime(2))^prime(1), therefore 675 is a term.
		

Links

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

Crossrefs

Intersection of A056166 and A076610.

Cf. A006450, A302590, A302596, A321874.

Programs

Mathematica

Join[{1}, Select[Range[35000], AllTrue[Join[PrimePi[(t = Transpose @ FactorInteger[#])[[1]]], t[[2]]], PrimeQ] &]] (* Amiram Eldar, Jul 30 2021 *)

Python

from sympy import factorint, isprime, primepi
def ok(n):
    f = factorint(n)
    if not all(isprime(e) for e in f.values()): return False
    return all(isprime(primepi(p)) for p in f)
print(list(filter(ok, range(35000)))) # Michael S. Branicky, Jul 30 2021

Formula

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + Sum_{q prime} 1/p^q) = 1.2271874... - Amiram Eldar, Jul 31 2021