A356323 -id:A356323 - cp's OEIS frontend

A356297 a(n) = n! * Sum_{k=1..n} sigma_0(k)/k.

1, 4, 16, 82, 458, 3228, 24036, 212448, 2032992, 21781440, 246853440, 3201742080, 42580650240, 621037186560, 9664270963200, 161166707251200, 2781679603046400, 52204357423411200, 1004687538456268800, 20823621371578368000, 447027656835852288000

Offset: 1

Seiichi Manyama, Aug 03 2022

nonn

Cf. A000005, A006218, A356010, A356298, A356323.

Table[n! * Sum[DivisorSigma[0, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)

PARI
```
a(n) = n!*sum(k=1, n, sigma(k, 0)/k);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k)/k)/(1-x)))

E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - x^k)/k.

a(n) ~ n! * (log(n)^2/2 + 2*gamma*log(n) + gamma^2 - 2*sg1), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Aug 07 2022

A356298 a(n) = n! * Sum_{k=1..n} sigma_2(k)/k.

Original entry on oeis.org

1, 7, 41, 290, 2074, 18444, 165108, 1749264, 19412496, 241299360, 3097006560, 45546606720, 673536159360, 10986261431040, 187460277177600, 3445281394329600, 64637392771123200, 1325310849663897600, 27498565425087590400, 616389533324974080000

Offset: 1

Seiichi Manyama, Aug 03 2022

nonn

Cf. A001157, A064602, A356010, A356297, A356323.

Table[n! * Sum[DivisorSigma[2, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)

PARI
```
a(n) = n!*sum(k=1, n, sigma(k, 2)/k);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k*(1-x^k)^2))/(1-x)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k*log(1-x^k))/(1-x)))

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k * (1 - x^k)^2).
E.g.f.: -(1/(1-x)) * Sum_{k>0} k * log(1 - x^k).
a(n) ~ n! * zeta(3) * n^2 / 2. - Vaclav Kotesovec, Aug 07 2022

A356485 a(n) = n! * Sum_{k=1..n} A000010(k)/k.

Original entry on oeis.org

1, 3, 13, 64, 416, 2736, 23472, 207936, 2113344, 22584960, 284722560, 3576337920, 52240412160, 768727895040, 12228344755200, 206114911027200, 3838718125670400, 71231050830643200, 1468632692485324800, 30345814652977152000, 666456931810639872000, 15172961921551171584000

Offset: 1

Vaclav Kotesovec, Aug 09 2022

nonn

Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A002088, A011755, A356297, A356010, A356298, A356323.

Table[n! * Sum[EulerPhi[k]/k, {k, 1, n}], {n, 1, 25}]

a(n) = n!*sum(k=1, n, eulerphi(k)/k); \\ Michel Marcus, Aug 09 2022

a(n) ~ n! * 6*n/Pi^2.

cp's OEIS Frontend

A356297 a(n) = n! * Sum_{k=1..n} sigma_0(k)/k.

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A356298 a(n) = n! * Sum_{k=1..n} sigma_2(k)/k.

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Formula

A356485 a(n) = n! * Sum_{k=1..n} A000010(k)/k.

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Mathematica

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Formula