A017665 Numerator of sum of reciprocals of divisors of n.

1, 3, 4, 7, 6, 2, 8, 15, 13, 9, 12, 7, 14, 12, 8, 31, 18, 13, 20, 21, 32, 18, 24, 5, 31, 21, 40, 2, 30, 12, 32, 63, 16, 27, 48, 91, 38, 30, 56, 9, 42, 16, 44, 21, 26, 36, 48, 31, 57, 93, 24, 49, 54, 20, 72, 15, 80, 45, 60, 14, 62, 48, 104, 127, 84, 24, 68, 63, 32, 72, 72, 65, 74, 57

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• nice

Comments

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Numerators of coefficients in expansion of Sum_{n >= 1} x^n / (n*(1-x^n)) = Sum_{n >= 1} log(1/(1-x^n)).

The primes in this sequence, in order of appearance (without multiplicity), begin: 3, 7, 2, 13, 31, 5, 127. The first occurrence of prime(k) = a(n) for k = 1, 2, 3, ... is at n = 6, 2, 24, 4, 35640, 9, 297600, 588, ... - Jonathan Vos Post, Apr 02 2011

With amicable numbers, we have a(A002025(n)) = a(A002046(n)). - Michel Marcus, Dec 29 2013

Numerator of sigma(n)/n = A000203(n)/n. See A239578(n) - the smallest number k such that a(k) = n. - Jaroslav Krizek, Sep 23 2014

Examples

			1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, 12/7, 8/5, 31/16, ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Paul A. Weiner, The abundancy ratio, a measure of perfection, Math. Mag. 73 (4) (2000) 307-310.
• Eric Weisstein's World of Mathematics, Abundancy.

Crossrefs

Cf. A000203, A002025, A002046, A013661, A013954-A013972, A017666 (denominators), A027750, A239578.

Programs

Haskell

import Data.Ratio ((%), numerator)
a017665 = numerator . sum . map (1 %) . a027750_row
-- Reinhard Zumkeller, Apr 06 2012

Magma

[Numerator(DivisorSigma(1,n)/n): n in [1..50]]; // G. C. Greubel, Nov 08 2018

Maple

with(numtheory): seq(numer(sigma(n)/n), n=1..74) ; # Zerinvary Lajos, Jun 04 2008

Mathematica

Numerator[DivisorSigma[-1,Range[80]]] (* Harvey P. Dale, May 31 2013 *)
Table[Numerator[DivisorSigma[1, n]/n], {n, 1, 50}] (* G. C. Greubel, Nov 08 2018 *)

PARI

a(n)=sigma(n)/gcd(n, sigma(n)) \\ Charles R Greathouse IV, Feb 11 2011

PARI

a(n)=numerator(sigma(n,-1)) \\ Charles R Greathouse IV, Apr 04 2011

Python

from math import gcd
from sympy import divisor_sigma
def A017665(n): return (m:=divisor_sigma(n))//gcd(m,n) # Chai Wah Wu, Mar 20 2023

Formula

a(n) = sigma(n)/gcd(n, sigma(n)). - Jon Perry, Jun 29 2003

Dirichlet g.f.: zeta(s)*zeta(s+1) [for fraction A017665/A017666]. - Franklin T. Adams-Watters, Sep 11 2005

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017666(k) = Pi^2/6 (A013661). - Amiram Eldar, Nov 21 2022