A372793 Sequence related to the asymptotic expansion of Sum_{k=1..n} tau(m*k).

1, 2, 3, 16, 5, 864, 7, 4096, 729, 64000, 11, 6879707136, 13, 2809856, 61509375, 4294967296, 17, 812479653347328, 19, 26843545600000000, 26795786661, 2791309312, 23, 4019988717840603673710821376, 9765625, 73719087104, 7625597484987, 25962355635465062711296, 29

Offset: 1

Author

Vaclav Kotesovec, May 13 2024

Keywords

• nonn

Comments

For m>=1, Sum_{k=1..n} tau(m*k) = A018804(m) * n * log(n) + O(n).

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.

Examples

			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.
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.
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.
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.
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.
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.
		

Crossrefs

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).

Cf. A018804, A193679.

Formula

Sum_{k=1..n} tau(m*k) ~ A018804(m) * n * (log(n) - 1 + 2*gamma)/m + n*log(a(m))/m.

a(m) = exp(limit_{n->oo} (m * (Sum_{k=1..n} tau(m*k)) - A018804(m)*n*(log(n) - 1 + 2*gamma))/n).

If p is prime, then a(p) = p.

If p is prime, then a(p^k) = p^(k*p^(k-1)).

If p and q are distinct primes, then a(p*q) = p^(2*q-1) * q^(2*p-1).