A352420 Number of distinct prime factors of sigma_n(n).

0, 1, 2, 3, 3, 4, 3, 2, 3, 5, 6, 8, 5, 5, 8, 6, 3, 8, 5, 11, 9, 7, 8, 10, 8, 8, 10, 12, 7, 13, 7, 11, 15, 10, 15, 11, 7, 8, 11, 10, 6, 14, 8, 14, 14, 11, 10, 17, 6, 21, 15, 16, 8, 18, 16, 15, 16, 6, 9, 22, 8, 10, 17, 13, 17, 17, 7, 17, 20, 17, 8, 23, 4, 13, 21

Offset: 1

Author

Wesley Ivan Hurt, Mar 21 2022

Keywords

• nonn

Examples

			a(5) = 3; a(5) = omega(sigma_5(5)) = omega(1^5+5^5) = omega(3126) = 3.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 1..120

Crossrefs

Cf. A001221 (omega), A023887 (sigma_n(n)).

Cf. A064165, A347718.

Programs

Maple

A342420 := proc(n)
    A001221(A023887(n)) ; # reuses other codes
end proc:
seq(A342420(n),n=1..20) ; # R. J. Mathar, Apr 06 2022

Mathematica

Table[PrimeNu[DivisorSigma[n, n]], {n, 30}]

PARI

a(n) = omega(sigma(n, n)); \\ Daniel Suteu, Mar 23 2022

Python

from sympy import primefactors, factorint
def A352420(n): return len(set().union(*(primefactors((p**((e+1)*n)-1)//(p**n-1)) for p, e in factorint(n).items()))) # Chai Wah Wu, Mar 24 2022

Formula

a(n) = omega(sigma_n(n)) = A001221(A023887(n)).

Extensions

a(67)-a(75) from Daniel Suteu, Mar 23 2022