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A131605
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Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).
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10
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1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624
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OFFSET
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1,2
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COMMENTS
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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, ...])
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LINKS
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PROG
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(PARI) isok(n) = if (n == 1, return (1), return (ispower(n, , &np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013
(Python)
from sympy import mobius, integer_nthroot, primepi
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())))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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Klaus Brockhaus previously provided a table of n, a(n) for n=1..1323, May 28 2008
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STATUS
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approved
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