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%I #15 May 13 2024 12:21:30
%S 1,2,3,16,5,864,7,4096,729,64000,11,6879707136,13,2809856,61509375,
%T 4294967296,17,812479653347328,19,26843545600000000,26795786661,
%U 2791309312,23,4019988717840603673710821376,9765625,73719087104,7625597484987,25962355635465062711296,29
%N Sequence related to the asymptotic expansion of Sum_{k=1..n} tau(m*k).
%C For m>=1, Sum_{k=1..n} tau(m*k) = A018804(m) * n * log(n) + O(n).
%C If p is prime, then Sum_{k=1..n} tau(p*k) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.
%F Sum_{k=1..n} tau(m*k) ~ A018804(m) * n * (log(n) - 1 + 2*gamma)/m + n*log(a(m))/m.
%F a(m) = exp(limit_{n->oo} (m * (Sum_{k=1..n} tau(m*k)) - A018804(m)*n*(log(n) - 1 + 2*gamma))/n).
%F If p is prime, then a(p) = p.
%F If p is prime, then a(p^k) = p^(k*p^(k-1)).
%F If p and q are distinct primes, then a(p*q) = p^(2*q-1) * q^(2*p-1).
%e Sum_{k=1..n} tau(4*k) ~ (8*n*(log(n) + 2*gamma - 1) + n*4*log(2)) / 4, a(4) = exp(4*log(2)) = 16.
%e Sum_{k=1..n} tau(6*k) ~ (15*n*(log(n) + 2*gamma - 1) + n*(5*log(2) + 3*log(3))) / 6, a(6) = exp(5*log(2) + 3*log(3)) = 864.
%e Sum_{k=1..n} tau(8*k) ~ (20*n*(log(n) + 2*gamma - 1) + n*12*log(2)) / 8, a(8) = exp(12*log(2)) = 4096.
%e Sum_{k=1..n} tau(9*k) ~ (21*n*(log(n) + 2*gamma - 1) + n*6*log(3)) / 9, a(9) = exp(6*log(3)) = 729.
%e Sum_{k=1..n} tau(10*k) ~ (27*n*(log(n) + 2*gamma - 1) + n*(9*log(2) + 3*log(5))) / 10, a(10) = exp(9*log(2) + 3*log(5)) = 64000.
%e Sum_{k=1..n} tau(12*k) ~ (40*n*(log(n) + 2*gamma - 1) + n*(20*log(2) + 8*log(3))) / 12, a(12) = exp(20*log(2) + 8*log(3)) = 6879707136.
%Y Cf. A000005 (m=1), A099777 (m=2), A372713 (m=3), A372784 (m=4), A372785 (m=5), A372786 (m=6), A372787 (m=7), A372788 (m=8), A372789 (m=9), A372790 (m=10), A372791 (m=11), A372792 (m=12).
%Y Cf. A018804, A193679.
%K nonn
%O 1,2
%A _Vaclav Kotesovec_, May 13 2024
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