A017665 - OEIS

%I #94 Feb 16 2025 08:32:33

%S 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,

%T 40,2,30,12,32,63,16,27,48,91,38,30,56,9,42,16,44,21,26,36,48,31,57,

%U 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

%N Numerator of sum of reciprocals of divisors of n.

%C 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

%C Numerators of coefficients in expansion of Sum_{n >= 1} x^n / (n*(1-x^n)) = Sum_{n >= 1} log(1/(1-x^n)).

%C 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

%C With amicable numbers, we have a(A002025(n)) = a(A002046(n)). - _Michel Marcus_, Dec 29 2013

%C 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

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.

%H T. D. Noe, <a href="/A017665/b017665.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul A. Weiner, <a href="http://www.jstor.org/stable/2690980">The abundancy ratio, a measure of perfection</a>, Math. Mag. 73 (4) (2000) 307-310.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Abundancy.html">Abundancy</a>.

%F a(n) = sigma(n)/gcd(n, sigma(n)). - _Jon Perry_, Jun 29 2003

%F Dirichlet g.f.: zeta(s)*zeta(s+1) [for fraction A017665/A017666]. - _Franklin T. Adams-Watters_, Sep 11 2005

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017666(k) = Pi^2/6 (A013661). - _Amiram Eldar_, Nov 21 2022

%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(numer(sigma(n)/n), n=1..74) ; # _Zerinvary Lajos_, Jun 04 2008

%t Numerator[DivisorSigma[-1,Range[80]]] (* _Harvey P. Dale_, May 31 2013 *)

%t Table[Numerator[DivisorSigma[1, n]/n], {n, 1, 50}] (* _G. C. Greubel_, Nov 08 2018 *)

%o (PARI) a(n)=sigma(n)/gcd(n, sigma(n)) \\ _Charles R Greathouse IV_, Feb 11 2011

%o (PARI) a(n)=numerator(sigma(n,-1)) \\ _Charles R Greathouse IV_, Apr 04 2011

%o (Haskell)

%o import Data.Ratio ((%), numerator)

%o a017665 = numerator . sum . map (1 %) . a027750_row

%o -- _Reinhard Zumkeller_, Apr 06 2012

%o (Magma) [Numerator(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 A017665(n): return (m:=divisor_sigma(n))//gcd(m,n) # _Chai Wah Wu_, Mar 20 2023

%Y Cf. A000203, A002025, A002046, A013661, A013954-A013972, A017666 (denominators), A027750, A239578.

%K nonn,frac,nice

%O 1,2

%A _N. J. A. Sloane_