A390760 - OEIS

%I #38 Jan 13 2026 08:08:43

%S 9,15,21,25,27,33,35,39,49,51,55,57,65,69,77,81,85,87,91,93,95,105,

%T 111,115,119,121,123,125,129,133,141,143,145,155,159,161,165,169,177,

%U 183,185,187,195,201,203,205,209,213,215,217,219,221,231,235,237,243,247,249,253,255,259

%N Odd composite numbers that are squarefree or prime powers.

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

%C 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).

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

%H Charles Kusniec, <a href="https://doi.org/10.5281/zenodo.18046455">Composite Numbers Map for the OEIS</a>.

%F Intersection of A071904 and (A024556 disjoint union A244623).

%t Select[Range[9, 259, 2], Xor[SquareFreeQ[#], PrimePowerQ[#]] &] (* _Michael De Vlieger_, Nov 23 2025 *)

%o (Python)

%o from math import isqrt

%o from sympy import primepi, integer_nthroot, mobius

%o def A390760(n):

%o     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)))

%o     m, k = n+1, f(n+1)

%o     while m != k: m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Nov 25 2025

%Y Equals A071904 \ A360769.

%Y Odd terms of A388427.

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

%K nonn,easy

%O 1,1

%A _Charles Kusniec_, Nov 17 2025