

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
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OFFSET

1,2


COMMENTS

Sum_{ d divides n } 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665A017712 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), A001157A001160 (k=2,3,4,5), A013954A013972 for k = 6,7,...,24.  Ahmed Fares (ahmedfares(AT)mydeja.com), Apr 05 2001
Denominators of coefficients in expansion of Sum_{n >= 1} x^n/(n*(1x^n)) = Sum_{n >= 1} log(1/(1x^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 (multiplyperfect numbers), a(n) = 2 for numbers n in A159907 (numbers n with halfintegral 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



