cp's OEIS Frontend

A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.

%I A340674 #24 Jul 27 2025 08:55:00
%S A340674 4,9,16,25,36,49,81,100,121,169,196,225,256,289,361,441,484,529,625,
%T A340674 676,841,900,961,1089,1156,1225,1296,1369,1444,1521,1681,1764,1849,
%U A340674 2116,2209,2401,2601,2809,3025,3249,3364,3481,3721,3844,4225,4356,4489,4761,4900
%N A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.
%H A340674 Antti Karttunen, <a href="/A340674/b340674.txt">Table of n, a(n) for n = 1..20000</a>
%F A340674 Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2^k)/zeta(2^(k+1))-1) = 0.6018231854... - _Amiram Eldar_, Feb 08 2021
%t A340674 Select[Range[10^4], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && u[[1]] == 2^IntegerExponent[u[[1]], 2] &] (* _Amiram Eldar_, Feb 08 2021 *)
%o A340674 (PARI)
%o A340674 A209229(n) = (n && !bitand(n,n-1));
%o A340674 isA340674(n) = { my(b,e); (((e=ispower(n,,&b))>0)&&issquarefree(b)&&A209229(e)); };
%o A340674 (Python)
%o A340674 from math import isqrt
%o A340674 from sympy import mobius, integer_nthroot
%o A340674 def A340674(n):
%o A340674     def bisection(f,kmin=0,kmax=1):
%o A340674         while f(kmax) > kmax: kmax <<= 1
%o A340674         kmin = kmax >> 1
%o A340674         while kmax-kmin > 1:
%o A340674             kmid = kmax+kmin>>1
%o A340674             if f(kmid) <= kmid:
%o A340674                 kmax = kmid
%o A340674             else:
%o A340674                 kmin = kmid
%o A340674         return kmax
%o A340674     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A340674     def f(x): return n+x-sum(g(integer_nthroot(x,1<<e)[0])-1 for e in range(1,x.bit_length().bit_length()))
%o A340674     return bisection(f,n,n) # _Chai Wah Wu_, Jun 01 2025
%Y A340674 Cf. A000079, A005117, A050376, A209229, A302777.
%Y A340674 Positions of terms larger than 2 in A340673 (also in A340675), and of terms larger than 1 in A340676.
%Y A340674 Subsequence of A072777 and of A340682.
%K A340674 nonn
%O A340674 1,1
%A A340674 _Antti Karttunen_, Jan 31 2021