A055225 -id:A055225 - cp's OEIS frontend

A356436 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d) )/k.

1, 5, 23, 146, 874, 8124, 62628, 707664, 7860816, 103284000, 1179669600, 24454569600, 324615427200, 5740203974400, 119579523436800, 2688723275212800, 46084905896601600, 1383333631684300800, 26411386476116275200, 868104140064602112000

Offset: 1

Seiichi Manyama, Aug 07 2022

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A055225, A353992, A356297, A356437, A356439.

a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d))/k);

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

a(n) = n! * Sum_{k=1..n} A055225(k)/k.
E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - k*x^k)/k.
a(n) ~ (n-1)! * 3^((n + 3 - mod(n,3))/3)/2. - Vaclav Kotesovec, Aug 07 2022

A308593 a(n) = Sum_{d|n} d^(n^2/d).

Original entry on oeis.org

1, 5, 28, 513, 3126, 840242, 823544, 8606711809, 7625984905477, 1221277338483250, 285311670612, 89215914432866222355906, 302875106592254, 316913110043605007120962336162, 608295209422788113565012727970423808, 680564733921105089459460296530789924865

Offset: 1

Seiichi Manyama, Jun 09 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..80

Diagonal of A308509.

Cf. A023887, A055225, A308571.

Table[Sum[d^(n^2/d), {d, Divisors[n]}], {n,1,20}] (* Vaclav Kotesovec, Jun 09 2019 *)

PARI
```
{a(n) = sumdiv(n, d, d^(n^2/d))}
```

A308675 a(n) = Sum_{d|n} d^(d^2 * n).

Original entry on oeis.org

1, 257, 7625597484988, 340282366920938463463374607431768276993, 2350988701644575015937473074444491355637331113544175043017503412556834518909454345703126

Offset: 1

Seiichi Manyama, Jun 16 2019

nonn

The next term has 169 digits. - Harvey P. Dale, Feb 29 2020

Column k=3 of A308676.

Cf. A023887, A055225, A308670.

Table[Total[#^(#^2 n)&/@Divisors[n]],{n,5}] (* Harvey P. Dale, Feb 29 2020 *)
a[n_] := DivisorSum[n, #^(n * #^2) &]; Array[a, 5] (* Amiram Eldar, May 11 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^(d^2*n))}
```

N=10; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k^k^2*x)^k)^(1/k)))))

L.g.f.: -log(Product_{k>=1} (1 - (k^(k^2)*x)^k)^(1/k)) = Sum_{k>=1} a(k)*x^k/k.

A344061 a(n) = Sum_{d|n} sigma(d)^(n/d).

Original entry on oeis.org

1, 4, 5, 17, 7, 56, 9, 146, 78, 298, 13, 1501, 15, 2276, 1265, 9219, 19, 25716, 21, 77519, 16929, 177328, 25, 739582, 7808, 1594562, 264382, 5611241, 31, 15699452, 33, 48863172, 4196081, 129140542, 312753, 447589422, 39, 1162261928, 67111665, 3771805472, 43, 10764897556, 45

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A007429, A055225, A279789, A309369, A342471, A344060.

a[n_] := DivisorSum[n, DivisorSigma[1 , #]^(n/#) &]; Array[a, 43] (* Amiram Eldar, May 08 2021 *)

PARI
```
a(n) = sumdiv(n, d, sigma(d)^(n/d));
```

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)*x^k/(1-sigma(k)*x^k)))

G.f.: Sum_{k >= 1} sigma(k) * x^k/(1 - sigma(k) * x^k).

If p is prime, a(p) = 2 + p.

A356588 Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k)^(1/k) )^x.

Original entry on oeis.org

1, 0, 2, 9, 44, 450, 2754, 45360, 340304, 6481944, 81801000, 1370631240, 21731534472, 511117017840, 8113055559504, 193958323289640, 4765385232157440, 108183734293844160, 2754467397591689664, 80416694712647352960, 2132862160676063137920, 67803682111729108433280

Offset: 0

Seiichi Manyama, Aug 14 2022

nonn

Cf. A055225, A355064, A356439, A356587.

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

a055225(n) = sumdiv(n, d, d^(n/d));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a055225(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * A055225(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A376016 a(n) = Sum_{d|n} d^(n/d) * binomial(n/d,d).

Original entry on oeis.org

1, 2, 3, 8, 5, 30, 7, 104, 36, 330, 11, 1296, 13, 2702, 2445, 7440, 17, 33030, 19, 51220, 76566, 112662, 23, 699216, 3150, 639002, 1653399, 2064412, 29, 10620300, 31, 12451872, 29229288, 17825826, 1640660, 190101888, 37, 89653286, 455976417, 441305440, 41

Offset: 1

Seiichi Manyama, Sep 06 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A055225, A327238, A376020.

a(n) = sumdiv(n, d, d^(n/d)*binomial(n/d, d));

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, (k*x^k)^k/(1-k*x^k)^(k+1)))

from math import comb
from itertools import takewhile
from sympy import divisors
def A376016(n): return sum(d**(m:=n//d)*comb(m,d) for d in takewhile(lambda d:d**2<=n,divisors(n))) # Chai Wah Wu, Sep 06 2024

G.f.: Sum_{k>=1} (k*x^k)^k / (1 - k*x^k)^(k+1).

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

A214845 Triangle read by rows: T(n,m) =(n/k)^(k-1) mod k, where k is the m-th divisor of n, 1 <= m <= tau(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 3, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 4, 1, 0, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 4, 3, 8, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 3, 1, 2, 1, 0, 1, 0, 1, 1, 1, 5, 3, 4, 1

Offset: 1

Gerasimov Sergey, Mar 08 2013

Row lengths are tau(n) = A000005(n).

The sequence of row sums starts: 0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 3, 3, 1, 1, 9, 1, 5, 3, 3, 1, 17, 1, 3, 1, 7, 1, 16, 1, 1, 3, 3, 3, 26, 1, 3, 3, 19, 1, 12, 1, 7, 18, 3, 1, 27, 1, 23...

			Triangle begins:
0;
0,1;
0,1;
0,0,1;
0,1;
0,1,1,1;
0,1;
0,0,0,1;
0,0,1;
0,1,1,1;
0,1;
0,0,1,3,2,1;
0,1;
0,1,1,1;
0,1,1,1;
0,0,0,0,1;
0,1;
0,1,0,3,4,1;

Cf. A000005, A027750, A055225, A087909, A213919.

A214845 := proc(n,m)
    sort(convert(numtheory[divisors](n),list)) ;
    k := op(m,%) ;
    modp((n/k)^(k-1),k) ;
end proc:
for n from 1 to 30 do
    for m from 1 to numtheory[tau](n) do
        printf("%d,",A214845(n,m)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Apr 17 2013

A308366 Expansion of Sum_{k>=1} (-1)^(k+1)kx^k/(1 - k*x^k).

Original entry on oeis.org

1, -1, 4, -7, 6, -4, 8, -39, 37, -16, 12, -94, 14, -92, 384, -591, 18, 65, 20, -1542, 2552, -1948, 24, -3606, 3151, -8048, 20440, -30590, 30, 33326, 32, -135455, 178512, -130816, 94968, -35029, 38, -523964, 1596560, -1749734, 42, 2521186, 44, -8374494, 16364502

Offset: 1

Ilya Gutkovskiy, May 22 2019

sign

Cf. A002129, A055225, A076717.

nmax = 45; CoefficientList[Series[Sum[(-1)^(k + 1) k x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 45; CoefficientList[Series[-Log[Product[(1 - k x^k)^((-1)^(k + 1)/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
Table[Sum[(-1)^(d + 1) d^(n/d), {d, Divisors[n]}], {n, 1, 45}]

L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^((-1)^(k+1)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(d+1)*d^(n/d).
a(n) = n + 1 if n is odd prime.

A343982 Numbers k that divide Sum_{j|k} j^(k/j).

Original entry on oeis.org

1, 6, 54, 135, 486, 495, 516, 1134, 1863, 2295, 3375, 4374, 4875, 5535, 10935, 11875, 15435, 19695, 22295, 23625, 24057, 34853, 39015, 39366, 42875, 43875, 59265, 64881, 77625, 84375, 89667, 100875, 102375, 114582, 122625, 142155, 144495, 161325, 165375, 205979, 251505, 268569

Offset: 1

Seiichi Manyama, May 06 2021

nonn

			1^6 + 2^3 + 3^2 + 6^1 = 24 = 4 * 6. So 6 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..200

Cf. A007691, A055225, A067313, A336892.

q[n_] := Divisible[DivisorSum[n, #^(n/#) &], n]; Select[Range[10^5], q] (* Amiram Eldar, May 06 2021 *)

isok(n) = sumdiv(n, d, Mod(d, n)^(n/d))==0;

A354851 a(n) = (n-1)! * Sum_{d|n} d^(n/d).

Original entry on oeis.org

1, 3, 8, 54, 144, 2880, 5760, 206640, 1491840, 24675840, 43545600, 10298534400, 6706022400, 1195587993600, 33476463820800, 775450900224000, 376610217984000, 553805325545472000, 128047474114560000, 339876410542276608000, 6208765924866785280000

Offset: 1

Seiichi Manyama, Jun 08 2022

nonn

Cf. A055225, A318249, A354848.

a[n_] := (n - 1)! * DivisorSum[n, #^(n/#) &]; Array[a, 20] (* Amiram Eldar, Jun 08 2022 *)

PARI
```
a(n) = (n-1)!*sumdiv(n, d, d^(n/d));
```

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

a(n) = (n-1)! * A055225(n).
E.g.f.: -Sum_{k>0} log(1 - k * x^k)/k.
If p is prime, a(p) = (p-1)! + p!.

cp's OEIS Frontend

A356436 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d) )/k.

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A308593 a(n) = Sum_{d|n} d^(n^2/d).

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A308675 a(n) = Sum_{d|n} d^(d^2 * n).

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PARI

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A344061 a(n) = Sum_{d|n} sigma(d)^(n/d).

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A356588 Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k)^(1/k) )^x.

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PARI

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A376016 a(n) = Sum_{d|n} d^(n/d) * binomial(n/d,d).

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PARI

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A214845 Triangle read by rows: T(n,m) =(n/k)^(k-1) mod k, where k is the m-th divisor of n, 1 <= m <= tau(n).

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A308366 Expansion of Sum_{k>=1} (-1)^(k+1)*k*x^k/(1 - k*x^k).

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Mathematica

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A343982 Numbers k that divide Sum_{j|k} j^(k/j).

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A308366 Expansion of Sum_{k>=1} (-1)^(k+1)kx^k/(1 - k*x^k).