A372674 a(n) = Sum_{j=1..n} Sum_{k=1..n} tau(j*k).

1, 8, 23, 54, 89, 162, 221, 326, 439, 596, 707, 964, 1107, 1352, 1645, 1976, 2179, 2630, 2865, 3390, 3859, 4316, 4615, 5406, 5883, 6444, 7059, 7892, 8299, 9430, 9877, 10794, 11635, 12424, 13361, 14852, 15415, 16324, 17349, 18952, 19587, 21342, 22017, 23486, 25177

Offset: 1

Author

Vaclav Kotesovec, May 10 2024

Keywords

• nonn

Comments

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

If p is prime, then Sum_{j=1..n} tau(p*j) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Plot of a(n) / (n^2 * (log(n) + 2*gamma - 1/2)^2) for n = 1..100000

Crossrefs

Cf. A372633, A372675.

Cf. A006218, A263086.

cf. A000005, A099777, A372713.

Programs

Mathematica

Table[Sum[DivisorSigma[0, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[0, n^2] + 2*Sum[DivisorSigma[0, j*n], {j, 1, n - 1}], {n, 2, 50}]]