A308047 - OEIS

%I #28 Sep 08 2022 08:46:21

%S 3,11,18,35,38,74,66,115,117,166,146,266,198,298,308,403,326,533,402,

%T 630,564,682,578,970,713,934,900,1162,902,1444,1026,1491,1316,1558,

%U 1396,2093,1446,1930,1812,2390,1766,2692,1938,2730,2522,2794,2306,3658,2565,3441

%N Sum of subgroup indices of dihedral group, Sum_{H <= D(n)} [D(n):H].

%C This is a generalization of the sum of divisors to finite groups:

%C Let G be a finite group. sigma(G) = Sum_{H <= G} [G:H]. For G = D(n) = dihedral group with 2*n elements, we get a(n) = 2*n * Sum_{H <= G} 1/|H|.

%C Relation to Lagarias inequality: Let H(G) be the harmonic numbers of D(n) relative to a generating set S <= D(n). Then the conjecture (Lagarias inequality) is:

%C    a(n) <= H(D(n)) + exp(H(D(n)))*log(H(D(n))).

%C Relation to sum of divisors: sigma(n) = sigma(C_n), where C_n = cyclic group.

%H MathStackexchange, <a href="https://math.stackexchange.com/questions/3224207/two-questions-about-the-dihedral-group"> Two questions about the dihedral group</a>, Proof of the formula a(n) = sigma_2(n) + 2*sigma_1(n).

%H MathOverflow, <a href="https://mathoverflow.net/questions/330077/a-group-theoretic-interpretation-of-lagarias-inequality"> A Group theoretic interpretation of Lagarias inequality</a>, Definition of the Harmonic numbers H(G) for each finite group G.

%H MathOverflow, <a href="https://mathoverflow.net/a/330599/6671"> Conjectured upper bound to number of subgroups</a>, Definition of sigma(G)=sigma_1 for each finite group G.

%F a(n) = 2*n * Sum_{H <= G} 1/|H|, where the sum runs through all subgroups H of G.

%F a(n) = sigma_2(n) + 2*sigma_1(n).

%e a(1) = 3 as there are two subgroups of order 1 and 2 so 2*1*(1/1 + 1/2) = 3.

%t Table[DivisorSigma[2,n] + 2*DivisorSigma[1,n], {n, 50}] (* _G. C. Greubel_, Jul 15 2019 *)

%o (Sage)

%o [sigma(n,2)+2*sigma(n) for n in range(1,51)]

%o (PARI) a(n) = sigma(n, 2) + 2*sigma(n); \\ _Michel Marcus_, May 13 2019

%o (Magma) [DivisorSigma(2,n) + 2*DivisorSigma(1,n): n in [1..50]]; // _G. C. Greubel_, Jul 15 2019

%K nonn

%O 1,1

%A _Orges Leka_, May 10 2019