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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)).

%I A265708 #24 Mar 27 2025 21:31:45
%S A265708 1,4,5,31,7,20,9,162,69,28,13,155,15,36,35,5127,19,276,21,217,45,52,
%T A265708 25,810,223,60,703,279,31,140,33,15536,65,76,63,2139,39,84,75,1134,43,
%U A265708 180,45,403,483,100,49,25635,521,892,95,465,55,2812,91,1458,105,124
%N 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)).
%H A265708 Antti Karttunen, <a href="/A265708/b265708.txt">Table of n, a(n) for n = 1..16384</a>
%F A265708 a(n) = A069934(n) * Sum_{d|n} 1/A000203(d) = A265709(n) * A069934(n) / A265710(n).
%F A265708 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))}.
%e A265708 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.
%t A265708 a[n_] := LCM @@ DivisorSigma[1, Divisors[n]] * DivisorSum[n, 1/DivisorSigma[1, #] &]; Array[a, 100] (* _Amiram Eldar_, Dec 09 2022 *)
%o A265708 (Magma) [&+[1/SumOfDivisors(d): d in Divisors(n)] * LCM([SumOfDivisors(d): d in Divisors(n)]): n in [1..100]];
%o A265708 (PARI)
%o A265708 A069934(n) = my(d = divisors(n)); lcm(vector(#d, k, sigma(d[k])));
%o A265708 A265708(n) = (A069934(n) * sumdiv(n,d,1/sigma(d))); \\ _Antti Karttunen_, Nov 19 2017
%Y A265708 Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265713, A265714, A266227, A266228.
%K A265708 nonn,mult
%O A265708 1,2
%A A265708 _Jaroslav Krizek_, Dec 24 2015