A377433 Number of non-perfect-powers x in the range prime(n) < x < prime(n+1).

0, 0, 1, 1, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 4, 5, 1, 4, 3, 1, 5, 2, 5, 7, 2, 1, 3, 1, 3, 11, 2, 5, 1, 8, 1, 5, 5, 3, 4, 5, 1, 9, 1, 2, 1, 11, 10, 2, 1, 3, 5, 1, 8, 4, 5, 5, 1, 5, 3, 1, 8, 13, 3, 1, 3, 12, 5, 8, 1, 3, 5, 6, 5, 5, 3, 5, 7, 2, 7, 9, 1, 9, 1, 5, 2

Offset

1,6

Comments

Non-perfect-powers (A007916) are numbers without a proper integer root.

Positions of terms > 1 appear to be A049579.

Links

Table of n, a(n) for n=1..85.

Formula

a(n) + A377432(n) = A046933(n) = prime(n+1) - prime(n) - 1.

Example

Between prime(4) = 7 and prime(5) = 11 the only non-perfect-power is 10, so a(4) = 1.

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1], radQ]], {n, 100}]

Prog

(Python)
from sympy import mobius, integer_nthroot, prime, nextprime
def A377433(n):
    def f(x): return int(x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    return -f(p:=prime(n))+f(nextprime(p)-1) # Chai Wah Wu, Dec 05 2025

Crossrefs

Positions of 1 are latter terms of A029707.

Positions of terms > 1 appear to be A049579.

For prime-powers instead of non-perfect-powers we have A080101.

For non-prime-powers instead of non-perfect-powers we have A368748.

Perfect-powers in the same range are counted by A377432.

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.

A065514 gives the greatest prime-power < prime(n), difference A377289.

A081676 gives the greatest perfect-power <= n.

A246655 lists the prime-powers not including 1, complement A361102.

A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.

A377468 gives the least perfect-power > n.

Cf. A000015, A002808, A024619, A031218, A064113, A065890, A244508, A377282, A377434, A377436, A377466, A377467.

Sequence in context: A162910 A098975 A309213 * A371103 A127121 A238703

Adjacent sequences: A377430 A377431 A377432 * A377434 A377435 A377436

Keyword

nonn

Author

Gus Wiseman, Nov 02 2024

Status

approved