A082522 Numbers of the form p^(2^k) with p prime and k>0.

4, 9, 16, 25, 49, 81, 121, 169, 256, 289, 361, 529, 625, 841, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 16129, 17161, 18769, 19321, 22201, 22801

Offset: 1

Author

Reinhard Zumkeller, May 11 2003

Keywords

• nonn

Comments

Every positive square (A000290 without 0) is the product of a unique subset of these numbers. The lexicographically earliest (when ordered) minimal set of generators for the positive squares as a group under A059897(.,.); the intersection of A050376 and A000290. - Peter Munn, Aug 25 2019

Examples

			3^(2^2) = 81, therefore 81 is a term.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Group.
• Eric Weisstein's World of Mathematics, Prime Power.
• Wikipedia, Generating set of a group.

Crossrefs

A050376 without A000040.

Cf. A000290, A000961, A025475, A059897.

Programs

PARI

lst(lim)=my(v=List(apply(n->n^2,primes(primepi(sqrtint(lim))))), t);forprime(p=2,(lim+.5)^(1/4),t=p^2;while((t=t^2)<=lim,listput(v,t)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Apr 10 2012

Python

from sympy import primepi, integer_nthroot
def A082522(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x,1<Chai Wah Wu, Feb 19 2025

Formula

a(n) = A050376(A181970(n)) = A050376(n)^2. - Vladimir Shevelev, Apr 05 2013

a(n) ~ n^2 log^2 n. - Charles R Greathouse IV, Oct 19 2015

Sum_{n>=1} 1/a(n) = Sum_{k>=1} P(2^k) = 0.53331724743088069672..., where P is the prime zeta function. - Amiram Eldar, Nov 26 2020