A367452 Number of semiprime divisors of the n-th squarefree number (A005117).

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 1, 0, 1, 1, 0, 3, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 3, 0, 1, 3, 0, 0, 1, 1, 3, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 3, 0, 3, 1, 0, 0, 3, 1, 0, 3, 1, 1, 1, 1, 1, 0, 1, 3, 0, 1, 1, 0, 3, 0, 1, 1, 1, 1, 1, 0, 0, 3, 1, 0, 1

Offset

1,19

Links

Table of n, a(n) for n=1..97.

Wikipedia, Squarefree integer.

Formula

a(n) = A086971(A005117(n)).

a(n) = c*(c-1)/2, where c = A001221(A005117(n)).

Example

a(19) = 3 since A005117(19) = 30 and 30 has 3 semiprime divisors, namely {6, 10, 15}.

Mathematica

Table[If[PrimeNu[n] == PrimeOmega[n], PrimeNu[n] (PrimeNu[n] - 1)/2, {}], {n, 200}] // Flatten

Prog

(PARI) apply(x->(sumdiv(x, d, bigomega(d)==2)), select(issquarefree, [1..300])) \\ Michel Marcus, Nov 22 2023
(Python)
from math import isqrt
from sympy import mobius, primenu
def A367452(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return (c:=primenu(m))*(c-1)>>1 # Chai Wah Wu, Aug 12 2024

Crossrefs

Cf. A001221 (omega), A001358 (semiprimes), A005117 (squarefree numbers), A086971.

Sequence in context: A036860 A119624 A213727 * A119612 A101949 A124796

Adjacent sequences: A367449 A367450 A367451 * A367453 A367454 A367455

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 18 2023

Status

approved