A058060 Number of distinct prime factors of d(n), the number of divisors of n.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset

1,12

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 9, 122, 1285, 13096, 131729, 1319621, 13203252, 132055132, 1320621032, 13206429426, 132064984784, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Rieger (1972) and Heppner (1974) (see the Formula section), can be empirically evaluated by 1.3206... . - Amiram Eldar, Jan 15 2024

References

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5001

E. Heppner, Über die Iteration von Teilerfunktionen, Journal für die reine und angewandte Mathematik, Vol. 265 (1974), pp. 176-182.

G. J. Rieger, Über einige arithmetische Summen, Manuscripta Mathematica, Vol. 7 (1972), pp. 23-34.

Formula

a(n) = A001221(A000005(n)).

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^5), where c is a constant (Rieger, 1972; Heppner, 1974). - Amiram Eldar, Jan 15 2024

Example

n = 120 = 8*3*5, d(n) = 16 = 2^4, so a(120)=1.

Mathematica

Table[PrimeNu[DivisorSigma[0, n]], {n, 1, 100}] (* G. C. Greubel, May 05 2017 *)

Prog

(PARI) a(n)=omega(numdiv(n)) \\ Charles R Greathouse IV, May 05 2017

Crossrefs

Cf. A001221, A000005, A058061.

Sequence in context: A294883 A348940 A088530 * A338160 A336137 A371735

Adjacent sequences: A058057 A058058 A058059 * A058061 A058062 A058063

Keyword

nonn,easy

Author

Labos Elemer, Nov 23 2000

Extensions

Offset corrected by Sean A. Irvine, Jul 22 2022

Status

approved