A377044 The n-th perfect-power A001597(n) minus the n-th prime-power A246655(n).

-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

Author

Gus Wiseman, Oct 25 2024

Keywords

• sign

Comments

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

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.

Cf. A023055, A045542, A052410, A053707, A069623, A110969, A216765, A376560, A376561, A377051.

Programs

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]

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

Formula

a(n) = A001597(n) - A246655(n).