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

%I A082522 #41 Feb 19 2025 12:34:30
%S A082522 4,9,16,25,49,81,121,169,256,289,361,529,625,841,961,1369,1681,1849,
%T A082522 2209,2401,2809,3481,3721,4489,5041,5329,6241,6561,6889,7921,9409,
%U A082522 10201,10609,11449,11881,12769,14641,16129,17161,18769,19321,22201,22801
%N A082522 Numbers of the form p^(2^k) with p prime and k>0.
%C A082522 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
%H A082522 Charles R Greathouse IV, <a href="/A082522/b082522.txt">Table of n, a(n) for n = 1..10000</a>
%H A082522 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Group.html">Group</a>.
%H A082522 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.
%H A082522 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>.
%F A082522 a(n) = A050376(A181970(n)) = A050376(n)^2. - _Vladimir Shevelev_, Apr 05 2013
%F A082522 a(n) ~ n^2 log^2 n. - _Charles R Greathouse IV_, Oct 19 2015
%F A082522 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
%e A082522 3^(2^2) = 81, therefore 81 is a term.
%o A082522 (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
%o A082522 (Python)
%o A082522 from sympy import primepi, integer_nthroot
%o A082522 def A082522(n):
%o A082522     def bisection(f,kmin=0,kmax=1):
%o A082522         while f(kmax) > kmax: kmax <<= 1
%o A082522         kmin = kmax >> 1
%o A082522         while kmax-kmin > 1:
%o A082522             kmid = kmax+kmin>>1
%o A082522             if f(kmid) <= kmid:
%o A082522                 kmax = kmid
%o A082522             else:
%o A082522                 kmin = kmid
%o A082522         return kmax
%o A082522     def f(x): return n+x-sum(primepi(integer_nthroot(x,1<<i)[0]) for i in range(1,x.bit_length().bit_length()))
%o A082522     return bisection(f,n,n) # _Chai Wah Wu_, Feb 19 2025
%Y A082522 A050376 without A000040.
%Y A082522 Cf. A000290, A000961, A025475, A059897.
%K A082522 nonn
%O A082522 1,1
%A A082522 _Reinhard Zumkeller_, May 11 2003