A265709 a(n) = numerator of Sum_{d|n} 1/sigma(d).

1, 4, 5, 31, 7, 5, 9, 54, 69, 14, 13, 155, 15, 3, 35, 1709, 19, 23, 21, 31, 45, 13, 25, 27, 223, 10, 703, 93, 31, 35, 33, 15536, 65, 38, 21, 713, 39, 7, 75, 9, 43, 15, 45, 403, 161, 25, 49, 1709, 521, 446, 95, 155, 55, 703, 91, 243, 21, 62, 61, 155, 63, 11

Offset

1,2

Comments

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

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

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

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

Example

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.

Mathematica

A265709[n_] := Numerator[DivisorSum[n, 1/DivisorSigma[1, #]&]];
Array[A265709, 100] (* Paolo Xausa, Feb 06 2024 *)

Prog

(Magma) [Numerator(&+[1/SumOfDivisors(d): d in Divisors(n)]): n in [1..1000]]
(PARI) A265709(n) = numerator(sumdiv(n, d, 1/sigma(d))); \\ Antti Karttunen, Nov 19 2017

Crossrefs

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

Sequence in context: A047169 A262034 A124482 * A265708 A224219 A128867

Adjacent sequences: A265706 A265707 A265708 * A265710 A265711 A265712

Keyword

nonn,frac

Author

Jaroslav Krizek, Dec 24 2015

Status

approved