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

Author

Reinhard Zumkeller, Apr 25 2003

Keywords

• nonn
• mult

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

Examples

			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.
		

Links

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

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).

Programs

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 *)

PARI

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

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

Extensions

Incorrect formula removed by Antti Karttunen, Jan 09 2023