A390760 Odd composite numbers that are squarefree or prime powers.

9, 15, 21, 25, 27, 33, 35, 39, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 105, 111, 115, 119, 121, 123, 125, 129, 133, 141, 143, 145, 155, 159, 161, 165, 169, 177, 183, 185, 187, 195, 201, 203, 205, 209, 213, 215, 217, 219, 221, 231, 235, 237, 243, 247, 249, 253, 255, 259

Offset

1,1

Comments

Odd composite numbers x whose prime factorization has either one prime squared or distinct primes.

Odd composite numbers x for which there exists y with 1 <= y <= floor(x/4) such that x*y is a square number (for odd prime-power x), or for which there exists y with 2 <= y <= floor(x/4) such that x*y is an oblong number (for odd squarefree x).

Omega(x) = omega(x) >= 2 (for odd squarefree composites), Omega(x) > omega(x) = 1 (for odd prime-power composites).

Links

Table of n, a(n) for n=1..61.

Charles Kusniec, Composite Numbers Map for the OEIS.

Formula

Intersection of A071904 and (A024556 disjoint union A244623).

Mathematica

Select[Range[9, 259, 2], Xor[SquareFreeQ[#], PrimePowerQ[#]] &] (* Michael De Vlieger, Nov 23 2025 *)

Prog

(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A390760(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, k)[0])-1 for k in range(2, x.bit_length()))+primepi(x)-sum(mobius(k)*(x//k**2+1>>1) for k in range(1, isqrt(x)+1, 2)))
    m, k = n+1, f(n+1)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Nov 25 2025

Crossrefs

Equals A071904 \ A360769.

Odd terms of A388427.

Cf. A024556 (odd squarefree composites), A244623 (odd prime-power composites).

Sequence in context: A155474 A100819 A046446 * A046350 A391475 A391474

Adjacent sequences: A390757 A390758 A390759 * A390761 A390762 A390763

Keyword

nonn,easy

Author

Charles Kusniec, Nov 17 2025

Status

approved