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

1,4

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

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.

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A292777 A088529 A267116 * A181591 A347442 A336424

Adjacent sequences: A136562 A136563 A136564 * A136566 A136567 A136568

Keyword

nonn

Author

Leroy Quet, Jan 07 2008

Extensions

More terms from Diana L. Mecum, Jul 17 2008

Status

approved