A372674 -id:A372674 - cp's OEIS frontend

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

1, 14, 59, 190, 401, 914, 1499, 2632, 4113, 6424, 8645, 13284, 17023, 23092, 30715, 40484, 48711, 63890, 75351, 95792, 116421, 139822, 159911, 199176, 229499, 267438, 309283, 364462, 404933, 482792, 532553, 611208, 688593, 772540, 862471, 998760, 1083615, 1200328

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^4 for n = 1..100000

Cf. A069097, A372633, A372674.
Cf. A024916, A326124.
Cf. A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.

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

a(n) ~ c * n^4, where c = Pi^4 / (144*zeta(3)) = 0.56274...

A372709 a(n) = Sum_{k=1..n} tau(n*k), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 5, 9, 18, 19, 41, 31, 56, 59, 83, 57, 136, 73, 127, 151, 170, 103, 233, 119, 270, 239, 233, 151, 406, 241, 285, 311, 424, 205, 579, 225, 464, 425, 399, 473, 758, 283, 459, 517, 812, 319, 891, 339, 742, 853, 587, 375, 1138, 571, 893, 731, 920, 437, 1157, 847, 1250

Offset: 1

Seiichi Manyama, May 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n) / A372674(n) for n = 1..100000

Cf. A000005, A048691, A263086, A372674, A372710.

Table[Sum[DivisorSigma[0, k*n], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, May 13 2024 *)

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

Conjecture: a(n) ~ A372674(n) / 2. - Vaclav Kotesovec, May 13 2024

cp's OEIS Frontend

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

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