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A377044
The n-th perfect-power A001597(n) minus the n-th prime-power A246655(n).
2
-1, 1, 4, 4, 9, 17, 18, 21, 23, 33, 47, 62, 77, 96, 98, 99, 113, 137, 159, 175, 182, 196, 207, 236, 265, 282, 297, 333, 370, 411, 433, 448, 493, 536, 579, 628, 681, 734, 791, 848, 879, 899, 962, 1028, 1094, 1159, 1192, 1220, 1293, 1364, 1437, 1514, 1559, 1591
OFFSET
1,3
COMMENTS
Perfect-powers (A001597) are numbers with a proper integer root.
FORMULA
a(n) = A001597(n) - A246655(n).
MATHEMATICA
perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All, 2]]>1;
per=Select[Range[1000], perpowQ];
per-NestList[NestWhile[#+1&, #+1, !PrimePowerQ[#]&]&, 2, Length[per]-1]
PROG
(Python)
from sympy import mobius, primepi, integer_nthroot
def A377044(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
def g(x): return int(n+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length())))
return bisection(f, n, n)-bisection(g, n, n) # Chai Wah Wu, Oct 27 2024
CROSSREFS
Including 1 with the prime-powers gives A377043.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, A093555, A376596.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A025475 lists numbers that are both a perfect-power and a prime-power.
A031218 gives the greatest prime-power <= n.
A080101 counts prime-powers between primes (exclusive).
A106543 lists numbers that are neither a perfect-power nor a prime-power.
A131605 lists perfect-powers that are not prime-powers.
A246655 lists the prime-powers, complement A361102, A375708.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Sequence in context: A059815 A202670 A203003 * A339427 A319646 A214826
KEYWORD
sign
AUTHOR
Gus Wiseman, Oct 25 2024
STATUS
approved