A372675 -id:A372675 - cp's OEIS frontend

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

Vaclav Kotesovec, May 10 2024

nonn

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.

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

Cf. A372633, A372675.
Cf. A006218, A263086.
cf. A000005, A099777, A372713.

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}]]

A372710 a(n) = Sum_{k=1..n} sigma(n*k).

Original entry on oeis.org

1, 10, 29, 81, 121, 302, 321, 630, 801, 1264, 1177, 2521, 1961, 3234, 4013, 5140, 4267, 8013, 5921, 10701, 10685, 12166, 10321, 20458, 15552, 19610, 21469, 28473, 20671, 40340, 25377, 40351, 39557, 43048, 45849, 70020, 43131, 59690, 63813, 89154, 58087, 106310

Offset: 1

Seiichi Manyama, May 11 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000203, A024916, A065764, A326124, A372675, A372709.

Table[Sum[DivisorSigma[1, k*n], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, May 13 2024 *)

PARI
```
a(n) = sum(k=1, n, sigma(n*k));
```

Conjecture: a(n) ~ A372675(n) / 2 ~ Pi^4 * n^4 / (288*zeta(3)). - Vaclav Kotesovec, May 13 2024

cp's OEIS Frontend

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

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