A062796 -id:A062796 - cp's OEIS frontend

A338694 a(n) = Sum_{d|n} d^d * binomial(d, n/d).

1, 8, 81, 1028, 15625, 280017, 5764801, 134219264, 3486784428, 100000031250, 3138428376721, 106993206079936, 3937376385699289, 155568095575106627, 6568408355712921875, 295147905179822588160, 14063084452067724991009, 708235345355351624428356, 37589973457545958193355601

Offset: 1

Seiichi Manyama, Apr 24 2021

nonn

Winston de Greef, Table of n, a(n) for n = 1..384

Cf. A062796, A318636, A318637, A318638, A338693.

a[n_] := DivisorSum[n, #^# * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)

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

N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+k*x^k)^k-k^k))

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

If p is prime, a(p) = p^(p+1).

A217576 a(n) = Sum_{d divides n} (d!)^(n/d).

Original entry on oeis.org

1, 3, 7, 29, 121, 765, 5041, 40913, 363097, 3643233, 39916801, 479535185, 6227020801, 87203692929, 1307676103777, 20924415922433, 355687428096001, 6402505760917569, 121645100408832001, 2432915176581403649, 51090942299733783937, 1124002321128529922049

Offset: 1

Joerg Arndt, Oct 07 2012

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..449

Cf. A062363 ( Sum_{d divides n} d! ).

Cf. A062796 ( Sum_{d divides n} d^d ), A066108 ( Sum_{d divides n} n^d ).

f[n_]=With[{d=Divisors[n]},Total[(d!)^(n/d)]]; Array[f,25] (* Harvey P. Dale, Dec 20 2023 *)

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

G.f.: Sum_{n>=1} n!*x^n / (1 - n!*x^n). - Paul D. Hanna, Jan 17 2013

A308670 a(n) = Sum_{d|n} d^(d*n).

Original entry on oeis.org

1, 17, 19684, 4294967553, 298023223876953126, 10314424798490535546559373642, 256923577521058878088611477224235621321608, 6277101735386680763835789423207666416120802188537744130049

Offset: 1

Seiichi Manyama, Jun 16 2019

nonn

Column k=2 of A308676.

Cf. A023887, A062796, A308571, A308671, A308675.

a[n_] := DivisorSum[n, #^(#*n) &]; Array[a, 8] (* Amiram Eldar, May 11 2021 *)

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

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k^k*x)^k)^(1/k)))))

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

A308674 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(d^k).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 17, 28, 7, 1, 257, 19684, 261, 6, 1, 65537, 7625597484988, 4294967313, 3126, 12, 1, 4294967297, 443426488243037769948249630619149892804, 340282366920938463463374607431768211713, 298023223876953126, 46688, 8

Offset: 1

Seiichi Manyama, Jun 16 2019

			Square array begins:
   1,   1,          1,                                       1, ...
   3,   5,         17,                                     257, ...
   4,  28,      19684,                           7625597484988, ...
   7, 261, 4294967313, 340282366920938463463374607431768211713, ...

Seiichi Manyama, Antidiagonals n = 1..9, flattened

Columns k=0..3 give A000203, A062796, A308671, A308672.

Cf. A308676.

T[n_, k_] := DivisorSum[n, #^(#^k) &]; Table[T[k, n - k], {n, 1, 7}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(j^k-1))).

A321385 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^d.

Original entry on oeis.org

1, 3, 28, 251, 3126, 46632, 823544, 16776955, 387420517, 9999996878, 285311670612, 8916100401824, 302875106592254, 11112006824734476, 437893890380862528, 18446744073692774139, 827240261886336764178, 39346408075296150201567, 1978419655660313589123980, 104857599999999989999997126

Offset: 1

Ilya Gutkovskiy, Nov 08 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A000312, A062796, A321387.

Table[Sum[(-1)^(n/d + 1) d^d, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[k^k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^d); \\ Michel Marcus, Nov 09 2018

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

a(n) ~ n^n. - Vaclav Kotesovec, Nov 09 2018

A345098 a(n) = Sum_{k=1..n} floor(n/k)^floor(n/k).

Original entry on oeis.org

1, 5, 29, 262, 3132, 46690, 823578, 16777484, 387420781, 10000003165, 285311673777, 8916100495209, 302875106639207, 11112006826381861, 437893890381686113, 18446744073726332260, 827240261886353544822, 39346408075296925042900

Offset: 1

Seiichi Manyama, Jun 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A062796, A345094, A345100.

a[n_] := Sum[Floor[n/k]^Floor[n/k], {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 08 2021 *)

PARI
```
a(n) = sum(k=1, n, (n\k)^(n\k));
```

my(N=20, x='x+O('x^N)); Vec(sum(j=1, N, (1-x^j)*sum(k=1, N, (k*x^j)^k))/(1-x))

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

a(n) ~ n^n. - Vaclav Kotesovec, Jun 11 2021

A345465 a(n) = Sum_{d|n} (d!)^d.

Original entry on oeis.org

1, 5, 217, 331781, 24883200001, 139314069504000221, 82606411253903523840000001, 6984964247141514123629140377600331781, 109110688415571316480344899355894085582848000000217, 395940866122425193243875570782668457763038822400000000024883200005

Offset: 1

Seiichi Manyama, Jul 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..30

Cf. A036739, A062796, A217576, A346196, A348146.

Total/@Table[((Divisors[n])!)^Divisors[n],{n,10}] (* Harvey P. Dale, Apr 24 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k!*x)^k/(1-x^k)))

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

If p is prime, a(p) = 1 + (p!)^p.

A308671 a(n) = Sum_{d|n} d^(d^2).

Original entry on oeis.org

1, 17, 19684, 4294967313, 298023223876953126, 10314424798490535546171968756, 256923577521058878088611477224235621321608, 6277101735386680763835789423207666416102355444468329480209

Offset: 1

Seiichi Manyama, Jun 16 2019

nonn

Column k=2 of A308674.

Cf. A000005, A062796, A308670.

a[n_] := DivisorSum[n, #^(#^2) &]; Array[a, 8] (* Amiram Eldar, May 11 2021 *)

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

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^(k^(k^2-1))))))

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

A308755 a(n) = Sum_{d|n} d^(d-2).

Original entry on oeis.org

1, 2, 4, 18, 126, 1301, 16808, 262162, 4782973, 100000127, 2357947692, 61917365541, 1792160394038, 56693912392105, 1946195068359504, 72057594038190098, 2862423051509815794, 121439531096599036046, 5480386857784802185940, 262144000000000100000143

Offset: 1

Seiichi Manyama, Jun 22 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A062796, A262843, A283498, A308753.

a[n_] := DivisorSum[n, #^(# - 2) &]; Array[a, 20] (* Amiram Eldar, May 08 2021 *)

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

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(k-3)))))

N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(k-2)*x^k/(1-x^k)))

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(k-3))) = Sum_{k>=1} a(k)*x^k/k.

G.f.: Sum_{k>=1} k^(k-2) * x^k/(1 - x^k).

A343983 Numbers k such that Sum_{j|k} j^j == 1 (mod k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 72, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Seiichi Manyama, May 06 2021

nonn

This sequence is different from A074583.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A062796, A188776.

q[n_] := Divisible[DivisorSum[n, #^# &] - 1, n]; Select[Range[260], q] (* Amiram Eldar, May 06 2021 *)

isok(n) = sumdiv(n, d, Mod(d, n)^d)==1;

from itertools import count, islice
from sympy import divisors
def A343983_gen(): # generator of terms
    yield 1
    for k in count(1):
        if sum(pow(j,j,k) for j in divisors(k,generator=True)) % k == 1:
            yield k
A343983_list = list(islice(A343983_gen(),30)) # Chai Wah Wu, Jun 19 2022

cp's OEIS Frontend

A338694 a(n) = Sum_{d|n} d^d * binomial(d, n/d).

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A217576 a(n) = Sum_{d divides n} (d!)^(n/d).

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A308670 a(n) = Sum_{d|n} d^(d*n).

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A308674 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(d^k).

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A321385 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^d.

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A345098 a(n) = Sum_{k=1..n} floor(n/k)^floor(n/k).

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A345465 a(n) = Sum_{d|n} (d!)^d.

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Mathematica

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A308671 a(n) = Sum_{d|n} d^(d^2).

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