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A017665
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Numerator of sum of reciprocals of divisors of n.
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78
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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. - comment from 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]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Abundancy
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FORMULA
| sigma(n)/gcd(n, sigma(n)) - Jon Perry, Jun 29 2003
Dirichlet generating function: zeta(s)*zeta(s+1) [for fraction A017665/A017666]. - Franklin T. Adams-Watters, Sep 11 2005.
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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, ...
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MAPLE
| with(numtheory): seq(numer(sigma(n)/n), n=1..74) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2008
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MATHEMATICA
| A017665[n_Integer] := Numerator[DivisorSigma[-1, n]]; A017665 /@ Range[100] (* From Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
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PROG
| (PARI) a(n)=sigma(n)/gcd(n, sigma(n)) \\ Charles R Greathouse IV, Feb 11 2011
a(n)=numerator(sigma(n, -1)) \\ Charles R Greathouse IV, Apr 04, 2011
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CROSSREFS
| Cf. A017666.
Sequence in context: A134688 A077650 A105853 * A105852 A190998 A067342
Adjacent sequences: A017662 A017663 A017664 * A017666 A017667 A017668
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KEYWORD
| nonn,frac,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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