A023887 -id:A023887 - cp's OEIS frontend

A259673 a(n) = sigma_(prime(n))(n).

1, 9, 244, 16513, 48828126, 13062296532, 232630513987208, 144115462954287105, 8862938119746644274757, 100000000186264514923632574038, 191943424957750480504146841291812, 8505622499882988712256991112913772434548, 4695452425098908797088971409337422035076128814

Offset: 1

Ilya Gutkovskiy, Jul 03 2015

Anders Hellström, Table of n, a(n) for n = 1..50
Index entries for sequences related to sums of divisors

Cf. A000203 (sigma(n)), A000040 (prime(n)), A023887 (sigma_n(n)).

Cf. A001157 (sigma_2), A001158 (sigma_3), A001160 (sigma_5), A013955 (sigma_7).

[DivisorSigma(NthPrime(n),n):n in [1..15]]; // Vincenzo Librandi, Jul 15 2015

a:= n-> numtheory[sigma][ithprime(n)](n):
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 10 2020

a[n_] := DivisorSigma[Prime[n], n]; Array[a, 13]
(* Second program: *)
a[n_] := SeriesCoefficient[Sum[k^Prime[n]*x^k/(1-x^k), {k, 1, n}], {x, 0, n}]; Array[a, 13] (* Jean-François Alcover, Sep 29 2017, from 2nd formula *)

a(n) = sigma(n, prime(n)); \\ Michel Marcus, Jul 03 2015

from sympy import divisor_sigma, prime
def A259673(n):
    return divisor_sigma(n,prime(n)) # Chai Wah Wu, Jul 20 2015

a(n) = sigma_(A000040(n))(n).

a(n) = [x^n] Sum_{k>=1} k^prime(n)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 26 2017

a(11) and a(12) from Anders Hellström, Jul 14 2015

A294948 Expansion of Product_{n>=1} (1 - n^n*x^n)^(1/n).

Original entry on oeis.org

1, -1, -2, -7, -57, -541, -7126, -108072, -1966034, -40620681, -952305757, -24824933859, -714742428220, -22491627743504, -768696164146118, -28344822040761041, -1121925480573229737, -47442205907345238412, -2134679753840086267669

Offset: 0

Seiichi Manyama, Nov 11 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/n, g(n) = n^n.

Seiichi Manyama, Table of n, a(n) for n = 0..387

Column k=0 of A294947.

Cf. A023881, A023887.

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

G.f.: exp(-Sum_{k>0} A023887(k)*x^k/k).

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A023887(k)*a(n-k) for n > 0.

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.

A321259 a(n) = sigma_n(n) - n^n.

Original entry on oeis.org

0, 1, 1, 17, 1, 794, 1, 65793, 19684, 9766650, 1, 2194095090, 1, 678223089234, 30531927033, 281479271743489, 1, 150196195641350171, 1, 100000096466944316978, 558545874543637211, 81402749386839765307626, 1, 79501574308536809523296482, 298023223876953126

Offset: 1

Ilya Gutkovskiy, Nov 01 2018

nonn

a(n) is the sum of n-th powers of proper divisors of n.

Seiichi Manyama, Table of n, a(n) for n = 1..400
Eric Weisstein's World of Mathematics, Proper divisors
Index entries for sequences related to sums of divisors

Cf. A000312, A001065, A023887, A067558, A276634, A279363, A279364, A292919, A321258.

[DivisorSigma(n, n) - n^n: n in [1..30]]; // Vincenzo Librandi, Nov 02 2018

Table[DivisorSigma[n, n] - n^n, {n, 25}]
nmax = 25; Rest[CoefficientList[Series[Sum[(k x)^(2 k)/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sigma(n, n) - n^n; \\ Michel Marcus, Nov 02 2018

G.f.: Sum_{k>=1} (k*x)^(2*k)/(1 - (k*x)^k).
a(n) = A023887(n) - A000312(n).
a(n) = A321258(n,n).
a(n) = 1 if n is prime.

A347718 a(n) = Sum of the divisors of sigma_n(n).

Original entry on oeis.org

1, 6, 56, 448, 6264, 96348, 1559520, 16908804, 391945400, 20553536052, 706019328000, 20210523379200, 519285252355776, 21710734431216480, 1456143373228677120, 25536237889612326912, 1792353900753729655758, 52839150354952425838080, 4154723599066412190910560

Offset: 1

Wesley Ivan Hurt, Jan 28 2022

nonn

			a(3) = sigma(sigma_3(3)) = sigma(1^3+3^3) = sigma(28) = 1+2+4+7+14+28 = 56.

Daniel Suteu, Table of n, a(n) for n = 1..120 (first 42 terms from Chai Wah Wu)
Index entries for sequences related to sigma(n)

Cf. A000203, A023887, A064165.

a:= n-> (s-> s(s[n](n)))(numtheory[sigma]):
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 28 2022

Table[DivisorSigma[1, DivisorSigma[n, n]], {n, 20}]

from math import prod
from collections import Counter
from sympy import factorint
def A347718(n): return prod((q**(r+1)-1)//(q-1) for q,r in sum((Counter(factorint((p**(n*(e+1))-1)//(p**n-1))) for p, e in factorint(n).items()),Counter()).items()) # Chai Wah Wu, Jan 28 2022

a(n) = sigma(sigma_n(n)).

a(n) = A000203(A023887(n)). - Michel Marcus, Jan 29 2022

A351749 a(n) = Sum_{p|n, p prime} sigma_p(p).

Original entry on oeis.org

0, 5, 28, 5, 3126, 33, 823544, 5, 28, 3131, 285311670612, 33, 302875106592254, 823549, 3154, 5, 827240261886336764178, 33, 1978419655660313589123980, 3131, 823572, 285311670617, 20880467999847912034355032910568, 33, 3126, 302875106592259

Offset: 1

Wesley Ivan Hurt, Feb 17 2022

nonn

Inverse Möbius transform of sigma_n(n) * c(n), where c(n) is the characteristic function of primes (A010051). - Wesley Ivan Hurt, Apr 01 2025

			a(6) = 33; a(6) = Sum_{p|6, p prime} sigma_p(p) = sigma_2(2) + sigma_3(3) = (1^2 + 2^2) + (1^3 + 3^3) = 33.

Cf. A010051, A023887 (sigma_n(n)).

a(p^k) = p^p + 1, for p prime and k >= 1.

a(n) = Sum_{d|n} sigma_d(d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 01 2025

A352841 Expansion of e.g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k/k!).

Original entry on oeis.org

1, 1, 7, 64, 851, 13906, 277972, 6466650, 172651643, 5186830537, 173327806752, 6373233407498, 255743444526584, 11119651415719744, 520752884139087852, 26132341317365562754, 1398900109763305183707, 79569524691656775423766

Offset: 0

Seiichi Manyama, Apr 05 2022

nonn

Cf. A340903, A340904, A352693.

Cf. A023887, A352839, A352843.

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

a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*binomial(n, k)*a(n-k)));

a(0) = 1; a(n) = Sum_{k=1..n} sigma_k(k) * binomial(n,k) * a(n-k).

A363662 a(n) = Sum_{d|n} (n/d)^n * binomial(d+n,n).

Original entry on oeis.org

2, 18, 128, 1590, 19002, 353304, 6591776, 154083654, 3878583770, 110647791078, 3423740752764, 116116072618104, 4240251502692142, 166761491097360240, 7006327371058071648, 313637735782416564806, 14890324713956395361406, 747610406539465959084870

Offset: 1

Seiichi Manyama, Jun 14 2023

nonn

Cf. A363646, A363647, A363648.

Cf. A023887, A363660, A363661.

a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + n, n] &]; Array[a, 20] (* Amiram Eldar, Jul 12 2023 *)

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

a(n) = [x^n] Sum_{k>0} (1/(1 - (k*x)^k)^(n+1) - 1).

A376015 a(n) = Sum_{d|n} d^n * binomial(n/d,d).

Original entry on oeis.org

1, 2, 3, 20, 5, 198, 7, 1544, 19692, 10250, 11, 2187216, 13, 344078, 143489085, 4296802320, 17, 7757846982, 19, 5497605324820, 366112362126, 230686742, 23, 4237941811999056, 298023223876953150, 5234491418, 640550188738935, 2522015815755104284

Offset: 1

Seiichi Manyama, Sep 06 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..724

Cf. A023887, A338685.

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

my(N=30, 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 A376015(n): return sum(d**n*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 - (k*x)^k)^(k+1).

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

cp's OEIS Frontend

A259673 a(n) = sigma_(prime(n))(n).

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A294948 Expansion of Product_{n>=1} (1 - n^n*x^n)^(1/n).

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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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A321259 a(n) = sigma_n(n) - n^n.

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A347718 a(n) = Sum of the divisors of sigma_n(n).

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A351749 a(n) = Sum_{p|n, p prime} sigma_p(p).

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A352841 Expansion of e.g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k/k!).

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