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A367452
Number of semiprime divisors of the n-th squarefree number (A005117).
1
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Nov 18 2023
STATUS
approved