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A017665
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Numerator of sum of reciprocals of divisors of n.
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156
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
P. A. Weiner, The abundancy ratio, a measure of perfection, Math. Mag. 73 (4) (2000) 307-310
Eric Weisstein's World of Mathematics, Abundancy
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FORMULA
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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
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EXAMPLE
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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
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with(numtheory): seq(numer(sigma(n)/n), n=1..74) ; # Zerinvary Lajos, Jun 04 2008
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000203, A002025, A002046, A013954-A013972, A017666, A027750, A239578.
Sequence in context: A105853 A277216 A323394 * A248789 A105852 A190998
Adjacent sequences: A017662 A017663 A017664 * A017666 A017667 A017668
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KEYWORD
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nonn,frac,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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