A083346 Denominator of r(n) = Sum(e/p: n=Product(p^e)).

1, 2, 3, 1, 5, 6, 7, 2, 3, 10, 11, 3, 13, 14, 15, 1, 17, 6, 19, 5, 21, 22, 23, 6, 5, 26, 1, 7, 29, 30, 31, 2, 33, 34, 35, 3, 37, 38, 39, 10, 41, 42, 43, 11, 15, 46, 47, 3, 7, 10, 51, 13, 53, 2, 55, 14, 57, 58, 59, 15, 61, 62, 21, 1, 65, 66, 67, 17, 69, 70, 71, 6, 73, 74, 15, 19, 77, 78

Offset

1,2

Comments

Multiplicative with a(p^e) = 1 iff p|e, p otherwise. For f(n) = A083345(n)/A083346(n), f(p^i*q^j*...) = f(p^i)+f(q^j)+ ... The denominator of each term is 1 or the prime, thus the denominator of the sum is the product of the denominators of the components. - Christian G. Bower, May 16 2005

n divided by the greatest common divisor of n and its arithmetic derivative, i.e., a(n) = n/gcd(n,n') = A000027(n)/A085731(n). - Giorgio Balzarotti, Apr 14 2011

Links

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

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 * Product_{p prime} (p^(2*p)*(p^2+p-1)-p^3)/((p^2+p-1)*(p^(2*p)-1)) = 0.3374565531... . - Amiram Eldar, Nov 18 2022

Example

n=12 = 2*2*3 = 2^2 * 3^1 -> r(12) = 2/2 + 1/3 = (6+2)/6, therefore a(12)=3, A083345(12)=4;
n=18 = 2*3*3 = 2^1 * 3^2 -> r(18) = 1/2 + 2/3 = (3+4)/6, therefore a(18)=6, A083345(18)=7.

Mathematica

a[n_] := Product[Module[{p, e}, {p, e} = pe; If[Divisible[e, p], 1, p]], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Oct 06 2021 *)

Prog

(PARI) A083346(n) = { my(f=factor(n)); denominator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); }; \\ Antti Karttunen, Mar 01 2018

Crossrefs

Cf. A083345 (numerator).

Cf. A035263 (parity of terms), A003159 (positions of odd terms), A036554 (of even terms).

Cf. A065463, A072873, A083347, A083348, A359588 (Dirichlet inverse).

Sequence in context: A072400 A007913 A366244 * A319652 A327938 A065883

Adjacent sequences: A083343 A083344 A083345 * A083347 A083348 A083349

Keyword

nonn,mult

Author

Reinhard Zumkeller, Apr 25 2003

Extensions

Incorrect formula removed by Antti Karttunen, Jan 09 2023

Status

approved