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A017665
Numerator of sum of reciprocals of divisors of n.
193
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
1,2
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
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.
LINKS
Paul A. Weiner, The abundancy ratio, a measure of perfection, Math. Mag. 73 (4) (2000) 307-310.
Eric Weisstein's World of Mathematics, Abundancy.
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
EXAMPLE
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, ...
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn,frac,nice
STATUS
approved