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 A017665 Numerator of sum of reciprocals of divisors of n. 144
 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; text; internal format)
 OFFSET 1,2 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. - 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 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 P. A. Weiner, The abundancy ratio, a measure of perfection, Math. Mag. 73 (4) (2000) 307-310 Eric Weisstein's World of Mathematics, Abundancy FORMULA 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 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(numer(sigma(n)/n), n=1..74) ; # Zerinvary Lajos, Jun 04 2008 MATHEMATICA 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 *) PROG (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 CROSSREFS 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 KEYWORD nonn,frac,nice AUTHOR STATUS approved

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Last modified October 22 16:26 EDT 2019. Contains 328318 sequences. (Running on oeis4.)