A265709 - OEIS

%I #15 Feb 06 2024 08:13:27

%S 1,4,5,31,7,5,9,54,69,14,13,155,15,3,35,1709,19,23,21,31,45,13,25,27,

%T 223,10,703,93,31,35,33,15536,65,38,21,713,39,7,75,9,43,15,45,403,161,

%U 25,49,1709,521,446,95,155,55,703,91,243,21,62,61,155,63,11

%N a(n) = numerator of Sum_{d|n} 1/sigma(d).

%C a(n) = numerator of Sum_{d|n} 1/A000203(d).

%C Are there numbers n > 1 such that Sum_{d|n} 1/sigma(d) is an integer?

%H Antti Karttunen, <a href="/A265709/b265709.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A265710(n) * Sum_{d|n} 1/sigma(d) = A265708(n) * A265710(n) / A069934(n).

%F a(1) = 1; a(p) = p + 2 for p = prime.

%e For n = 6; divisors d of 6: {1, 2, 3, 6}; sigma(d): {1, 3, 4, 12}; Sum_{d|6} 1/sigma(d) = 1/1 + 1/3 + 1/4 + 1/12 = 20/12 = 5/3; a(n) = 5.

%t A265709[n_] := Numerator[DivisorSum[n, 1/DivisorSigma[1,#]&]];

%t Array[A265709, 100] (* _Paolo Xausa_, Feb 06 2024 *)

%o (Magma) [Numerator(&+[1/SumOfDivisors(d): d in Divisors(n)]): n in [1..1000]]

%o (PARI) A265709(n) = numerator(sumdiv(n,d,1/sigma(d))); \\ _Antti Karttunen_, Nov 19 2017

%Y Cf. A069934, A000203, A265708, A265710, A265711, A265712, A265713, A265714, A266227, A266228.

%K nonn,frac

%O 1,2

%A _Jaroslav Krizek_, Dec 24 2015