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%I #51 Mar 21 2023 13:13:11
%S 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,
%T 29,5,31,32,11,17,35,36,37,19,39,4,41,7,43,11,15,23,47,12,49,50,17,26,
%U 53,9,55,7,57,29,59,5,61,31,63,64,65,11,67,34,23,35,71,24,73,37,75,19
%N Denominator of sum of reciprocals of divisors of n.
%C 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
%C Denominators of coefficients in expansion of Sum_{n >= 1} x^n/(n*(1-x^n)) = Sum_{n >= 1} log(1/(1-x^n)).
%C 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
%C a(A007691(n)) = 1. - _Reinhard Zumkeller_, Apr 06 2012
%C 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
%C For all n, a(n) <= n, and thus records are obtained for terms of A014567. - _Michel Marcus_, Sep 25 2014
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.
%H T. D. Noe, <a href="/A017666/b017666.txt">Table of n, a(n) for n=1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundancy.html">Abundancy</a>
%e 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, ...
%p with(numtheory): seq(denom(sigma(n)/n), n=1..76) ; # _Zerinvary Lajos_, Jun 04 2008
%t Table[Denominator[DivisorSigma[-1, n]], {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2011 *)
%t Table[Denominator[DivisorSigma[1, n]/n], {n, 1, 50}] (* _G. C. Greubel_, Nov 08 2018 *)
%o (Haskell)
%o import Data.Ratio ((%), denominator)
%o a017666 = denominator . sum . map (1 %) . a027750_row
%o -- _Reinhard Zumkeller_, Apr 06 2012
%o (PARI) a(n) = denominator(sigma(n)/n); \\ _Michel Marcus_, Sep 23 2014
%o (Magma) [Denominator(DivisorSigma(1,n)/n): n in [1..50]]; // _G. C. Greubel_, Nov 08 2018
%o (Python)
%o from math import gcd
%o from sympy import divisor_sigma
%o def A017666(n): return n//gcd(divisor_sigma(n),n) # _Chai Wah Wu_, Mar 21 2023
%Y Cf. A017665, A027750.
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Labos Elemer_, Aug 14 2002
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