A346010 a(n) is the denominator of the average number of distinct prime factors of the divisors of n.

1, 2, 2, 3, 2, 1, 2, 4, 3, 1, 2, 6, 2, 1, 1, 5, 2, 6, 2, 6, 1, 1, 2, 4, 3, 1, 4, 6, 2, 2, 2, 6, 1, 1, 1, 3, 2, 1, 1, 4, 2, 2, 2, 6, 6, 1, 2, 10, 3, 6, 1, 6, 2, 4, 1, 4, 1, 1, 2, 3, 2, 1, 6, 7, 1, 2, 2, 6, 1, 2, 2, 12, 2, 1, 6, 6, 1, 2, 2, 10, 5, 1, 2, 3, 1, 1

Offset: 1

Author

Amiram Eldar, Jul 01 2021

Keywords

• nonn
• frac

Comments

See A346009 for details.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• R. L. Duncan, Note on the divisors of a number, The American Mathematical Monthly, Vol. 68, No. 4 (1961), pp. 356-359.
• Sébastien Gaboury, Sur les convolutions de fonctions arithmétiques, M.Sc. thesis, Laval University, Quebec, 2007.

Crossrefs

Cf. A346009 (numerators).

Programs

Mathematica

a[n_] := Denominator[DivisorSum[n, PrimeNu[#] &]/DivisorSigma[0, n]]; Array[a, 100]
(* or *)
f[p_, e_] := e/(e+1); a[1] = 1; a[n_] := Denominator[Plus @@ f @@@ FactorInteger[n]]; Array[a, 100]