A131605 - OEIS

%I #19 Aug 14 2024 14:46:28

%S 1,36,100,144,196,216,225,324,400,441,484,576,676,784,900,1000,1089,

%T 1156,1225,1296,1444,1521,1600,1728,1764,1936,2025,2116,2304,2500,

%U 2601,2704,2744,2916,3025,3136,3249,3364,3375,3600,3844,3969,4225,4356,4624

%N Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

%C Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])

%H Daniel Forgues, <a href="/A131605/b131605.txt">Table of n, a(n) for n=1..8649</a>

%o (PARI) isok(n) = if (n == 1, return (1), return (ispower(n, ,&np) && (! isprime(np)))); \\ _Michel Marcus_, Jun 12 2013

%o (Python)

%o from sympy import mobius, integer_nthroot, primepi

%o def A131605(n):

%o     def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x,k)[0])-1)+primepi(a) for k in range(2,x.bit_length())))

%o     kmin, kmax = 1,2

%o     while f(kmax) >= kmax:

%o         kmax <<= 1

%o     while True:

%o         kmid = kmax+kmin>>1

%o         if f(kmid) < kmid:

%o             kmax = kmid

%o         else:

%o             kmin = kmid

%o         if kmax-kmin <= 1:

%o             break

%o     return kmax # _Chai Wah Wu_, Aug 14 2024

%Y Cf. A000961, A001597, A024619, A025475.

%K nonn

%O 1,2

%A _Daniel Forgues_, May 27 2008

%E _Klaus Brockhaus_ previously provided a table of n, a(n) for n=1..1323, May 28 2008