A265708 a(n) = lcm_{d|n} sigma(d) * Sum_{d|n} 1/sigma(d), where sigma(d) represents the sum of divisors of d (A000203(d)).

1, 4, 5, 31, 7, 20, 9, 162, 69, 28, 13, 155, 15, 36, 35, 5127, 19, 276, 21, 217, 45, 52, 25, 810, 223, 60, 703, 279, 31, 140, 33, 15536, 65, 76, 63, 2139, 39, 84, 75, 1134, 43, 180, 45, 403, 483, 100, 49, 25635, 521, 892, 95, 465, 55, 2812, 91, 1458, 105, 124

Offset: 1

Author

Jaroslav Krizek, Dec 24 2015

Keywords

• nonn
• mult

Examples

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

Links

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

Crossrefs

Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265713, A265714, A266227, A266228.

Programs

Magma

[&+[1/SumOfDivisors(d): d in Divisors(n)] * LCM([SumOfDivisors(d): d in Divisors(n)]): n in [1..100]];

Mathematica

a[n_] := LCM @@ DivisorSigma[1, Divisors[n]] * DivisorSum[n, 1/DivisorSigma[1, #] &]; Array[a, 100] (* Amiram Eldar, Dec 09 2022 *)

PARI

A069934(n) = my(d = divisors(n)); lcm(vector(#d, k, sigma(d[k])));
A265708(n) = (A069934(n) * sumdiv(n,d,1/sigma(d))); \\ Antti Karttunen, Nov 19 2017

Formula

a(n) = A069934(n) * Sum_{d|n} 1/A000203(d) = A265709(n) * A069934(n) / A265710(n).

Multiplicative with a(p^e) = (1/1 + ..., + 1/sigma(p^(e-1)) + 1/sigma(p^(e))) * lcm{1, ..., sigma(p^(e-1)), sigma(p^(e))}.