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A082522 Numbers of the form p^(2^k) with p prime and k>0. 6

%I #32 Nov 26 2020 03:18:42

%S 4,9,16,25,49,81,121,169,256,289,361,529,625,841,961,1369,1681,1849,

%T 2209,2401,2809,3481,3721,4489,5041,5329,6241,6561,6889,7921,9409,

%U 10201,10609,11449,11881,12769,14641,16129,17161,18769,19321,22201,22801

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

%C 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 Charles R Greathouse IV, <a href="/A082522/b082522.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Group.html">Group</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.

%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>.

%F a(n) = A050376(A181970(n)) = A050376(n)^2. - _Vladimir Shevelev_, Apr 05 2013

%F a(n) ~ n^2 log^2 n. - _Charles R Greathouse IV_, Oct 19 2015

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

%o (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

%Y A050376 without A000040.

%Y Cf. A000290, A000961, A025475, A059897.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, May 11 2003

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)