A083345 Numerator of r(n) = Sum(e/p: n=Product(p^e)).

0, 1, 1, 1, 1, 5, 1, 3, 2, 7, 1, 4, 1, 9, 8, 2, 1, 7, 1, 6, 10, 13, 1, 11, 2, 15, 1, 8, 1, 31, 1, 5, 14, 19, 12, 5, 1, 21, 16, 17, 1, 41, 1, 12, 13, 25, 1, 7, 2, 9, 20, 14, 1, 3, 16, 23, 22, 31, 1, 23, 1, 33, 17, 3, 18, 61, 1, 18, 26, 59, 1, 13, 1, 39, 11, 20, 18, 71, 1, 11, 4, 43, 1, 31, 22

Offset

1,6

Comments

Least common multiple of n and its arithmetic derivative, divided by n, i.e. a(n) = lcm(n,n')/n = A086130(n)/A000027(n). - Giorgio Balzarotti, Apr 14 2011

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

The fraction a(n)/A083346(n) is totally additive with a(p) = 1/p. - Franklin T. Adams-Watters, May 17 2006

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A083346(k) = Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141). - Amiram Eldar, Sep 29 2023

a(n) = A003415(n) / A085731(n) = A342001(n) / A369008(n). - Antti Karttunen, Jan 16 2024

Example

Fractions begin with 0, 1/2, 1/3, 1, 1/5, 5/6, 1/7, 3/2, 2/3, 7/10, 1/11, 4/3, ...
For n = 12, 2*2*3 = 2^2 * 3^1 --> r(12) = 2/2 + 1/3 = (6+2)/6, therefore a(12) = 4, A083346(12) = 3.
For n = 18, 2*3*3 = 2^1 * 3^2 --> r(18) = 1/2 + 2/3 = (3+4)/6, therefore a(18) = 7, A083346(18) = 6.

Mathematica

Array[Numerator@ Total[FactorInteger[#] /. {p_, e_} /; e > 0 :> e/p] - Boole[# == 1] &, 85] (* Michael De Vlieger, Feb 25 2018 *)

Prog

(PARI) A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); }; \\ Antti Karttunen, Feb 25 2018

Crossrefs

Cf. A083346 (denominator), A000027, A072873, A083347, A083348, A085731, A086130, A136141, A342001, A342002 [= a(A276086(n))].

Cf. A369002 (positions of even terms), A369003 (of odd terms), A369005 (of multiples of 4), A369007 (of multiples of 27), A369008, A369068 (Möbius transform), A369069.

Sequence in context: A215010 A136744 A068237 * A087262 A082343 A166125

Adjacent sequences: A083342 A083343 A083344 * A083346 A083347 A083348

Keyword

nonn,easy,frac

Author

Reinhard Zumkeller, Apr 25 2003

Status

approved