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A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.

%I A318750 #19 Jan 19 2025 09:35:34
%S A318750 1,7,16,40,55,109,130,210,264,354,387,603,642,768,903,1143,1194,1518,
%T A318750 1575,1935,2124,2322,2391,3111,3261,3495,3765,4269,4356,5166,5259,
%U A318750 5931,6228,6534,6849,8145,8256,8598,8949,10149,10272,11406,11535,12327,13137,13551
%N A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.
%H A318750 Vaclav Kotesovec, <a href="/A318750/b318750.txt">Table of n, a(n) for n = 1..100000</a>
%H A318750 Vaclav Kotesovec, <a href="/A318750/a318750_1.jpg">Graph - The asymptotic ratio (1000000 terms)</a>
%F A318750 a(n) = Sum_{k=1..n} A034718(k).
%F A318750 a(n) ~ n^2 * (log(n)^2 + (6*g-1)*log(n) + 6*g^2 - 3*g - 6*g1 + 1/2) / 4, where g is the Euler-Mascheroni constant A001620 and g1 is the first Stieltjes constant A082633. - _Vaclav Kotesovec_, Sep 09 2018
%t A318750 Accumulate[Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 100}]]
%t A318750 (* Asymptotics: *) n^2 * (Log[n]^2 + (6*EulerGamma - 1)*Log[n] + 6*EulerGamma^2 - 3*EulerGamma - 6*StieltjesGamma[1] + 1/2) / 4 (* _Vaclav Kotesovec_, Sep 09 2018 *)
%o A318750 (PARI) tau_3(k) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(k)[, 2]));
%o A318750 a(n) = sum(k = 1, n,  k * tau_3(k)); \\ _Amiram Eldar_, Jan 18 2025
%Y A318750 Cf. A007425, A034718, A061201, A318413, A318414.
%Y A318750 Cf. A001620, A082633.
%K A318750 nonn
%O A318750 1,2
%A A318750 _Vaclav Kotesovec_, Sep 02 2018