A377043 The n-th perfect-power A001597(n) minus the n-th power of a prime A000961(n).

0, 2, 5, 5, 11, 18, 19, 23, 25, 36, 48, 64, 81, 98, 100, 101, 115, 138, 164, 179, 184, 200, 209, 240, 271, 284, 300, 336, 374, 413, 439, 450, 495, 542, 587, 632, 683, 738, 793, 852, 887, 903, 964, 1029, 1097, 1165, 1194, 1230, 1295, 1370, 1443, 1518, 1561

Offset

1,2

Comments

Perfect-powers (A001597) are numbers with a proper integer root.

Links

Table of n, a(n) for n=1..53.

Formula

a(n) = A001597(n) - A000961(n).

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All, 2]]>1;
per=Select[Range[1000], perpowQ];
per-NestList[NestWhile[#+1&, #+1, !PrimePowerQ[#]&]&, 1, Length[per]-1]

Prog

(Python)
from sympy import mobius, primepi, integer_nthroot
def A377043(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-1+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

Excluding 1 from the powers of primes gives A377044.

A000015 gives the least prime-power >= n.

A031218 gives the greatest prime-power <= n.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

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.

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 (differences A375708).

Prime-power runs: A373675, min A373673, max A373674, length A174965.

Prime-power antiruns: A373576, min A120430, max A006549, length A373671.

Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A377051.

Sequence in context: A274108 A368616 A138316 * A183719 A379564 A239340

Adjacent sequences: A377040 A377041 A377042 * A377044 A377045 A377046

Keyword

nonn

Author

Gus Wiseman, Oct 25 2024

Status

approved