A062796 -id:A062796 - cp's OEIS frontend

A355887 a(n) = Sum_{k=1..n} k^k * floor(n/k).

1, 6, 34, 295, 3421, 50109, 873653, 17651130, 405071647, 10405074777, 295716745389, 9211817240589, 312086923832843, 11424093750214407, 449317984131076935, 18896062057857406028, 846136323944194170206, 40192544399241119212807

Offset: 1

Seiichi Manyama, Jul 20 2022

nonn

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

Partial sums of A062796.

Cf. A006218, A024916, A064602, A064603, A064604, A248076, A319194.

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

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

def A355887(n): return n*(1+n**(n-1))+sum(k**k*(n//k) for k in range(2,n)) if n>1 else 1 # Chai Wah Wu, Jul 21 2022

a(n) = Sum_{k=1..n} Sum_{d|k} d^d.

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

A359731 a(n) = (1/2) * Sum_{d|n} (2*d)^d.

Original entry on oeis.org

1, 9, 109, 2057, 50001, 1493109, 52706753, 2147485705, 99179645293, 5120000050009, 292159150705665, 18260173719523445, 1240576436601868289, 91029559915023973833, 7174453500000000050109, 604462909807316734838793, 54214017802982966177103873

Offset: 1

Seiichi Manyama, Jan 12 2023

Seiichi Manyama, Table of n, a(n) for n = 1..351

Cf. A062796, A076723, A359732.

a[n_] := DivisorSum[n, (2*#)^# &] / 2; Array[a, 20] (* Amiram Eldar, Aug 14 2023 *)

PARI
```
a(n) = sumdiv(n, d, (2*d)^d)/2;
```

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

G.f.: Sum_{k>0} 2^(k-1) * (k * x)^k / (1 - x^k).

A076723 Sum_{d divides n} (-d)^d.

Original entry on oeis.org

-1, 3, -28, 259, -3126, 46632, -823544, 16777475, -387420517, 9999996878, -285311670612, 8916100495144, -302875106592254, 11112006824734476, -437893890380862528, 18446744073726329091, -827240261886336764178, 39346408075296150201567

Offset: 1

Vladeta Jovovic, Oct 27 2002

sign

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

Cf. A062796, A321385.

a(n) = sumdiv(n, d, (-d)^d); \\ Michel Marcus, Dec 22 2018

G.f.: Sum_{n>0} (-n*x)^n/(1-x^n).

A343567 a(n) = Sum_{d|n} (n/d)^(n/d) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 6, 33, 292, 3195, 47154, 824467, 16783176, 387434574, 10000082730, 285311855367, 8916101760828, 302875109296409, 11112006847596746, 437893890421433595, 18446744074133995664, 827240261886937844567, 39346408075305857765940

Offset: 1

Seiichi Manyama, Apr 20 2021

nonn

Cf. A062796, A343547, A343568.

a[n_] := DivisorSum[n, (n/#)^(n/#) * Binomial[# + n - 2, n - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 25 2021 *)

a(n) = sumdiv(n, d, (n/d)^(n/d)*binomial(d+n-2, n-1));

a(n) = [x^n] Sum_{k>=1} (k * x)^k/(1 - x^k)^n.

A376014 a(n) = Sum_{d|n} d^d * binomial(n/d,d).

Original entry on oeis.org

1, 2, 3, 8, 5, 18, 7, 32, 36, 50, 11, 180, 13, 98, 285, 384, 17, 702, 19, 1480, 966, 242, 23, 5640, 3150, 338, 2295, 9352, 29, 22440, 31, 18432, 4488, 578, 65660, 85500, 37, 722, 7761, 229560, 41, 337302, 43, 85448, 406080, 1058, 47, 1449360, 823592, 788750, 18411

Offset: 1

Seiichi Manyama, Sep 06 2024

nonn

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

Cf. A062796, A338694, A376018.

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

my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, (k*x^k)^k/(1-x^k)^(k+1)))

from math import comb
from itertools import takewhile
from sympy import divisors
def A376014(n): return sum(d**d*comb(n//d,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 - x^k)^(k+1).

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

A308672 a(n) = Sum_{d|n} d^(d^3).

Original entry on oeis.org

1, 257, 7625597484988, 340282366920938463463374607431768211713, 2350988701644575015937473074444491355637331113544175043017503412556834518909454345703126

Offset: 1

Seiichi Manyama, Jun 16 2019

nonn

The next term (a(6)) has 169 digits. - Harvey P. Dale, Sep 08 2020

Column k=3 of A308674.

Cf. A000005, A062796, A308671.

Table[Total[Divisors[n]^Divisors[n]^3],{n,5}] (* Harvey P. Dale, Sep 08 2020 *)
a[n_] := DivisorSum[n, #^(#^3) &]; Array[a, 5] (* Amiram Eldar, May 11 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^d^3)}
```

N=10; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^(k^(k^3-1))))))

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

A336892 Numbers that are a divisor of the sum of their divisors to their own powers.

Original entry on oeis.org

1, 10, 12, 96, 304, 639, 2052, 2060, 2097, 2940, 5586, 9087, 10550, 38988, 42622, 84380, 128030, 199694, 255240, 342411, 346044, 515316, 673233, 721035, 1053700, 1361943, 2149875, 4206049, 5739687, 6979316, 10896431, 15904273, 138156772, 144608991, 276866005

Offset: 1

Scott R. Shannon, Aug 07 2020

nonn

As n is a divisor of n^n this sequence is also the numbers that are a divisor of the sum of their proper divisors to their own powers.
Integers k such that k divides A062796(k). - Michel Marcus, Aug 07 2020
a(36) > 280 million if it exists. - David A. Corneth, Aug 10 2020

			10 is a term as the divisors of 10 are 1,2,5,10 and 1^1+2^2+5^5+10^10 = 3130 + 10^10 which is divisible by 10.
12 is a term as the divisors of 12 are 1,2,3,4,6,12 and 1^1+2^2+3^3+4^4+6^6+12^12 = 46944 + 12^12 which is divisible by 12.

Cf. A032741, A000005, A001065, A032742, A062796.

seqQ[n_] := Divisible[DivisorSum[n, PowerMod[#, #, n] &], n]; Select[Range[10^5], seqQ] (* Amiram Eldar, Aug 10 2020 *)

isokb(k) = ! sumdiv(k, d, if (dMichel Marcus, Aug 10 2020

a(19)-a(25) from Amiram Eldar, Aug 08 2020
a(26)-a(32) from Michel Marcus, Aug 10 2020
a(33)-a(35) from David A. Corneth, Aug 10 2020

A347399 a(n) = A347398(n^n).

Original entry on oeis.org

1, 5, 28, 261, 3126, 46688, 823544, 16777477, 387420517, 10000003386, 285311670612, 8916117272416, 302875106592254, 11112006826381820, 437893890380862528, 18446744073726329093, 827240261886336764178, 39346408075296925042857

Offset: 1

Seiichi Manyama, Aug 30 2021

nonn

This sequence is different from A062796.

Cf. A000312, A062796, A347397, A347398.

PARI
```
a(n) = sum(k=1, n, (n^n%k^k==0)*k^k);
```

a(n) = A347397(n^n) - A347397(n^n-1) for n > 1.
a(n) = Sum_{k=1..n, k^k | n^n} k^k.
If p is prime, a(p) = 1 + p^p.

A351165 a(n) = n! * Sum_{d|n} d^(d - n/d).

Original entry on oeis.org

1, 6, 60, 1584, 75120, 5601960, 592956000, 84557864160, 15620794842240, 3628800457682400, 1035338990353113600, 355902198996315787200, 145077660657865961625600, 69194697633957032681544000, 38174841090323471644830720000, 24122334398251368151021076928000

Offset: 1

Seiichi Manyama, Aug 21 2022

nonn

Cf. A062796, A087905, A355669, A356661.

a[n_] := n! * DivisorSum[n, #^(# - n/#) &]; Array[a, 16] (* Amiram Eldar, Aug 21 2022 *)

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

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

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

A363590 a(n) = Sum_{d|n, d odd} d^d.

Original entry on oeis.org

1, 1, 28, 1, 3126, 28, 823544, 1, 387420517, 3126, 285311670612, 28, 302875106592254, 823544, 437893890380862528, 1, 827240261886336764178, 387420517, 1978419655660313589123980, 3126, 5842587018385982521381947992, 285311670612

Offset: 1

Seiichi Manyama, Jul 08 2023

Not multiplicative: a(3)*a(5) != a(15), for example.

Cf. A062796, A333823, A333824.

Cf. A000593, A050999, A051000, A292919, A363991.

a[n_] := DivisorSum[n, #^# &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Jul 26 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%2==1)*d^d);
```

from sympy import divisors
def A363590(n): return sum(d**d for d in divisors(n>>(~n & n-1).bit_length(),generator=True)) # Chai Wah Wu, Jul 09 2023

G.f.: Sum_{k>0} ((2*k-1) * x)^(2*k-1) / (1 - x^(2*k-1)).

a(2^n) = 1.

cp's OEIS Frontend

A355887 a(n) = Sum_{k=1..n} k^k * floor(n/k).

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A359731 a(n) = (1/2) * Sum_{d|n} (2*d)^d.

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A076723 Sum_{d divides n} (-d)^d.

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A343567 a(n) = Sum_{d|n} (n/d)^(n/d) * binomial(d+n-2,n-1).

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Mathematica

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Formula

A376014 a(n) = Sum_{d|n} d^d * binomial(n/d,d).

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Python

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A308672 a(n) = Sum_{d|n} d^(d^3).

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Formula

A336892 Numbers that are a divisor of the sum of their divisors to their own powers.

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PARI

Extensions

A347399 a(n) = A347398(n^n).

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