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

Author

Labos Elemer, Nov 23 2000

Keywords

• nonn
• easy

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

Examples

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

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.

Crossrefs

Cf. A001221, A000005, A058061.

Programs

Mathematica

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

PARI

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

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

Extensions

Offset corrected by Sean A. Irvine, Jul 22 2022