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A017666 Denominator of sum of reciprocals of divisors of n. 72
1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 6, 19, 10, 21, 11, 23, 2, 25, 13, 27, 1, 29, 5, 31, 32, 11, 17, 35, 36, 37, 19, 39, 4, 41, 7, 43, 11, 15, 23, 47, 12, 49, 50, 17, 26, 53, 9, 55, 7, 57, 29, 59, 5, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35, 71, 24, 73, 37, 75, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

Eric Weisstein's World of Mathematics, Abundancy

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A017665, A027750.

Sequence in context: A100994 A140523 A237517 * A253247 A201059 A319654

Adjacent sequences:  A017663 A017664 A017665 * A017667 A017668 A017669

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Labos Elemer, Aug 14 2002

STATUS

approved

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Last modified January 17 10:30 EST 2019. Contains 319218 sequences. (Running on oeis4.)