# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a320966 Showing 1-1 of 1 %I A320966 #35 Sep 15 2024 22:01:30 %S A320966 8,16,27,32,64,72,81,108,125,128,144,200,216,243,256,288,324,343,392, %T A320966 400,432,500,512,576,625,648,675,729,784,800,864,968,972,1000,1024, %U A320966 1125,1152,1296,1323,1331,1352,1372,1568,1600,1728,1800,1936,1944,2000,2025,2048,2187,2197,2304,2312,2401,2500 %N A320966 Powerful numbers A001694 divisible by a cube > 1. %C A320966 Powerful numbers that are not squares of squarefree numbers. - _Amiram Eldar_, Jun 25 2022 %H A320966 Hugo Pfoertner, Table of n, a(n) for n = 1..10000 %F A320966 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 15/Pi^2 = 0.4237786821... . - _Amiram Eldar_, Jun 25 2022 %t A320966 Select[Range[2500], (m = MinMax[FactorInteger[#][[;; , 2]]])[[1]] > 1 && m[[2]] > 2 &] (* _Amiram Eldar_, Jun 25 2022 *) %o A320966 (PARI) isA001694(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]==1, return(0))); 1 \\ from _Charles R Greathouse IV_ %o A320966 isA046099(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]>2, return(1)));0 %o A320966 for (k=1,2500,if(isA001694(k)&&isA046099(k),print1(k,", "))) %o A320966 (Python) %o A320966 from math import isqrt %o A320966 from sympy import mobius, integer_nthroot %o A320966 def A320966(n): %o A320966 def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1))) %o A320966 def bisection(f,kmin=0,kmax=1): %o A320966 while f(kmax) > kmax: kmax <<= 1 %o A320966 while kmax-kmin > 1: %o A320966 kmid = kmax+kmin>>1 %o A320966 if f(kmid) <= kmid: %o A320966 kmax = kmid %o A320966 else: %o A320966 kmin = kmid %o A320966 return kmax %o A320966 def f(x): %o A320966 c, l = n+x+squarefreepi(isqrt(x))-squarefreepi(integer_nthroot(x,3)[0]), 0 %o A320966 j = isqrt(x) %o A320966 while j>1: %o A320966 k2 = integer_nthroot(x//j**2,3)[0]+1 %o A320966 w = squarefreepi(k2-1) %o A320966 c -= j*(w-l) %o A320966 l, j = w, isqrt(x//k2**3) %o A320966 return c+l %o A320966 return bisection(f,n,n) # _Chai Wah Wu_, Sep 15 2024 %Y A320966 Cf. A000578, A320965. %Y A320966 Intersection of A001694 and A046099. %Y A320966 A001694 \ A062503. %K A320966 nonn %O A320966 1,1 %A A320966 _Hugo Pfoertner_, Oct 25 2018 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE