1,2

Sum_{ d divides 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

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

Also n/gcd(n, sigma(n)) = n/A009194(n); also n/lcm(all common divisors of n and sigma(n)). Equals 1 if 6,28,120,496,672,8128,..., i.e., if n is from A007691. - Labos Elemer, Aug 14 2002

a(A007691(n)) = 1. - Reinhard Zumkeller, Apr 06 2012

Denominator of sigma(n)/n = A000203(n)/n. a(n) = 1 for numbers n in A007691 (multiply-perfect numbers), a(n) = 2 for numbers n in A159907 (numbers n with half-integral abundancy index), a(n) = 3 for numbers n in A245775, a(n) = n for numbers n in A014567 (numbers n such that n and sigma(n) are relatively prime). See A162657 (n) - the smallest number k such that a(k) = n. - Jaroslav Krizek, Sep 23 2014

For all n, a(n) <= n, and thus records are obtained for terms of A014567. - Michel Marcus, Sep 25 2014

Conjecture: If a(n) is in A005153, then n is in A005153. In particular, if n has dyadic rational abundancy index, i.e., a(n) is in A000079 (such as A007691 and A159907), then n is in A005153. Since every term of A005153 greater than 1 is even, any odd n such that a(n) in A005153 must be in A007691. It is natural to ask if there exists a generalization of the indicator function for A005153, call it m(n), such that m(n) = 1 for n in A005153, 0 < m(n) < 1 otherwise, and m(a(n)) <= m(n) for all n. See also A050972. - Jaycob Coleman, Sep 27 2014

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

T. D. Noe, Table of n, a(n) for n=1..10000

Eric Weisstein's World of Mathematics, Abundancy

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, ...

with(numtheory): seq(denom(sigma(n)/n), n=1..76) ; # Zerinvary Lajos, Jun 04 2008

Table[Denominator[DivisorSigma[-1, n]], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)

Table[Denominator[DivisorSigma[1, n]/n], {n, 1, 50}] (* G. C. Greubel, Nov 08 2018 *)

(Haskell)

import Data.Ratio ((%), denominator)

a017666 = denominator . sum . map (1 %) . a027750_row

-- Reinhard Zumkeller, Apr 06 2012

(PARI) a(n) = denominator(sigma(n)/n); \\ Michel Marcus, Sep 23 2014

(Magma) [Denominator(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 A017666(n): return n//gcd(divisor_sigma(n), n) # Chai Wah Wu, Mar 21 2023

Cf. A017665, A027750.

Sequence in context: A140523 A237517 A332883 * A253247 A201059 A325959

Adjacent sequences: A017663 A017664 A017665 * A017667 A017668 A017669

nonn,frac

More terms from Labos Elemer, Aug 14 2002

approved