A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 4, 2, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 5, 2, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 3, 6, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 3, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 6, 1, 3, 3, 2, 1, 1, 1, 4, 1

Offset: 1

Author

Leroy Quet, Jan 07 2008

Keywords

• nonn

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 192, 2089, 21405, 215730, 2162136, 21636277, 216410510, 2164253043, 21642998932, ... . Apparently, the asymptotic mean of this sequence is 2.164... . - Amiram Eldar, Jun 30 2025

Examples

			120 = 2^3 * 3^1 * 5^1. The exponents of the prime factorization are therefore 3,1,1. The distinct values which equal these exponents are 1 and 3. So a(120) = 1+3 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Diana Mecum)
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A066328, A071625, A088529, A136566, A136568, A181591, A181819.

Programs

Mathematica

Join[{0},Table[Total[Union[Transpose[FactorInteger[n]][[2]]]],{n,2,110}]] (* Harvey P. Dale, Jun 23 2013 *)

PARI

A136565(n) = vecsum(apply(primepi,factor(factorback(apply(e->prime(e),(factor(n)[,2]))))[,1])); \\ Antti Karttunen, Sep 06 2018

Formula

a(n) = A088529(n) = A181591(n) for n: 2 <= n < 24. - Reinhard Zumkeller, Nov 01 2010

a(n) = A066328(A181819(n)). - Antti Karttunen, Sep 06 2018

Extensions

More terms from Diana L. Mecum, Jul 17 2008