A023887 -id:A023887 - cp's OEIS frontend

A157255 a(n) = sigma_(n^n)(n).

1, 17, 7625597484988

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 26 2009

Amiram Eldar, Table of n, a(n) for n = 1..4

Mathematica
```
Table[DivisorSigma[n^n,n],{n,1,5}]
```

a(n) = sigma(n, n^n); \\ Amiram Eldar, Mar 01 2024

A226890 E.g.f.: exp( Sum_{n>=1} sigma(n,n) * x^(n^2) / n^n ).

Original entry on oeis.org

1, 1, 1, 1, 31, 151, 451, 1051, 33601, 663601, 5187001, 25905001, 254322751, 10408719751, 128046088171, 920598820051, 29249420054401, 723848667813601, 12441294278905201, 138598703861148241, 4406639731521827551, 93453608310743628151, 1932981245635597160851, 27744052310106087405451

Offset: 0

Paul D. Hanna, Jun 20 2013

nonn

Here sigma(n,n) = A023887(n), the sum of the n-th powers of the divisors of n.

Compare to: exp( Sum_{n>=1} sigma(n)*x^n/n ), the g.f. of the partitions.

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 31*x^4/4! + 151*x^5/5! + 451*x^6/6! +...
where
log(A(x)) = x + 5*x^4/2^2 + 28*x^9/3^3 + 273*x^16/4^4 + 3126*x^25/5^5 + 47450*x^36/6^6 + 823544*x^49/7^7 +...+ A023887(n)*x^(n^2)/n^n +...

Cf. A226838, A023887.

{a(n)=n!*polcoeff(exp(sum(m=1,n,sigma(m,m)*(x^m/m)^m)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

a(n) == 1 (mod 30) (conjecture - valid up to n=4000; if true for n>=0, why?).

A308763 a(n) = Sum_{d|n} d^(n-2).

Original entry on oeis.org

1, 2, 4, 21, 126, 1394, 16808, 266305, 4785157, 100390882, 2357947692, 61978939050, 1792160394038, 56707753666594, 1946196290656824, 72061992352890881, 2862423051509815794, 121441386937936123331, 5480386857784802185940, 262145000003883417004506

Offset: 1

Seiichi Manyama, Jun 23 2019

nonn

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

Cf. A023887, A082245, A294645, A294810, A308755.

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

PARI
```
{a(n) = sigma(n, n-2)}
```

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

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

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

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

A319278 Square array sigma_k(n) read down antidiagonals: sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 7, 1, 17, 28, 21, 6, 1, 33, 82, 73, 26, 12, 1, 65, 244, 273, 126, 50, 8, 1, 129, 730, 1057, 626, 252, 50, 15, 1, 257, 2188, 4161, 3126, 1394, 344, 85, 13, 1, 513, 6562, 16513, 15626, 8052, 2402, 585, 91, 18, 1, 1025, 19684, 65793, 78126, 47450, 16808, 4369, 757, 130, 12

Offset: 1

R. J. Mathar, Sep 16 2018

Equals the square array A082771 without its first column.

			The array starts in row n=1 with columns k>=1 as:
     1      1      1      1      1      1       1        1
     3      5      9     17     33     65     129      257
     4     10     28     82    244    730    2188     6562
     7     21     73    273   1057   4161   16513    65793
     6     26    126    626   3126  15626   78126   390626
    12     50    252   1394   8052  47450  282252  1686434
     8     50    344   2402  16808 117650  823544  5764802
    15     85    585   4369  33825 266305 2113665 16843009

Columns: A000203, A001157, A001158, A001159.
Rows: A000012, A000051, A034472, A001576, A034474, A034488.
Cf. A082771, A023887 (diagonal), A109974, A319194 (partial column sums).

T[n_, k_] := DivisorSigma[k, n];
Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

sigma_k(n) = sum_{d|n} d^k.

A321190 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k/(1 - x^k)).

Original entry on oeis.org

1, 1, 6, 47, 778, 25476, 1752936, 242632397, 70015221566, 41446777283255, 49999934258165654, 125272856707074638221, 641938223803783115191706, 6731818441446626626586172740, 146378489075644780343627471981694, 6505906463580477520696075719916583118

Offset: 0

Ilya Gutkovskiy, Oct 29 2018

nonn

Cf. A023887, A129921, A180305, A319647, A320649, A321042.

seq(coeff(series((1-add(k^n*x^k/(1-x^k),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 29 2018

Table[SeriesCoefficient[1/(1 - Sum[k^n x^k/(1 - x^k), {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[DivisorSigma[n, k] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[Sum[j^n x^(i j), {j, 1, n}], {i, 1, n}]), {x, 0, n}], {n, 0, 15}]

a(n) = [x^n] 1/(1 - Sum_{k>=1} sigma_n(k)*x^k).
a(n) = [x^n] 1/(1 - Sum_{i>=1, j>=1} j^n*x^(i*j)).
a(n) = [x^n] 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^k)^(k^(n-1)))).

A321295 a(n) = n * sigma_n(n).

Original entry on oeis.org

1, 10, 84, 1092, 15630, 284700, 5764808, 134744072, 3486961557, 100097666500, 3138428376732, 107019534520152, 3937376385699302, 155577590681061500, 6568408813691796120, 295152408847700721680, 14063084452067724991026, 708238048886859220660710

Offset: 1

Ilya Gutkovskiy, Nov 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A023887, A064987.

[n*DivisorSigma(n, n): n in [1..20]]; // Vincenzo Librandi, Nov 06 2018

Table[n DivisorSigma[n, n], {n, 18}]
nmax = 18; Rest[CoefficientList[Series[Sum[k^(k + 1) x^k/(1 - (k x)^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[EulerPhi[n/d] DivisorSigma[n + 1, d], {d, Divisors[n]}], {n, 18}]

a(n) = n*sigma(n, n); \\ Michel Marcus, Nov 03 2018

use ntheory ":all"; say "$ ",vecprod($,divisor_sum($,$)) for 1..30; # Dana Jacobsen, Nov 05 2018

G.f.: Sum_{k>=1} k^(k+1)*x^k/(1 - (k*x)^k)^2.
a(n) = Sum_{d|n} phi(n/d)*sigma_(n+1)(d).
a(n) = n * A023887(n).

A346773 a(n) = Sum_{d|n} möbius(d)^n.

Original entry on oeis.org

1, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 2, 0, 8, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0

Offset: 1

Seiichi Manyama, Aug 03 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008683, A023887, A034444, A068068, A080323, A100008, A264782.

Table[Sum[MoebiusMu[d]^n,{d,Divisors[n]}],{n,103}] (* Stefano Spezia, Aug 03 2021 *)

PARI
```
a(n) = sumdiv(n, d, moebius(d)^n);
```
PARI
```
a(n) = if(n%2, 0^(n-1), 2^omega(2*n));
```

N=99; x='x+O('x^N); Vec(sum(k=1, N, (moebius(k)*x)^k/(1-(moebius(k)*x)^k)))

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

a(2*n-1) = 0^(n-1) and a(2*n) = A034444(2*n) = A100008(n) for n > 0.

A350503 Self-convolution of sigma_n(n).

Original entry on oeis.org

1, 10, 81, 826, 9766, 141448, 2371173, 46285454, 1025108476, 25583455958, 709014239914, 21645251877538, 721074846998772, 26036416881319860, 1012400617533469985, 42173042024213765280, 1873237518724869171647, 88367918629612594749582, 4411615292876287372479856

Offset: 1

Wesley Ivan Hurt, Jan 06 2022

nonn

Cf. A023887 (sigma_n(n)).

Table[Sum[DivisorSigma[k, k]*DivisorSigma[n - k + 1, (n - k + 1)], {k, n}], {n, 20}]

a(n) = Sum_{k=1..n} A023887(k) * A023887(n-k+1).

a(n) ~ 2*n^n. - Vaclav Kotesovec, Aug 20 2025

A352420 Number of distinct prime factors of sigma_n(n).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 3, 2, 3, 5, 6, 8, 5, 5, 8, 6, 3, 8, 5, 11, 9, 7, 8, 10, 8, 8, 10, 12, 7, 13, 7, 11, 15, 10, 15, 11, 7, 8, 11, 10, 6, 14, 8, 14, 14, 11, 10, 17, 6, 21, 15, 16, 8, 18, 16, 15, 16, 6, 9, 22, 8, 10, 17, 13, 17, 17, 7, 17, 20, 17, 8, 23, 4, 13, 21

Offset: 1

Wesley Ivan Hurt, Mar 21 2022

nonn

			a(5) = 3; a(5) = omega(sigma_5(5)) = omega(1^5+5^5) = omega(3126) = 3.

Daniel Suteu, Table of n, a(n) for n = 1..120

Cf. A001221 (omega), A023887 (sigma_n(n)).

Cf. A064165, A347718.

A342420 := proc(n)
    A001221(A023887(n)) ; # reuses other codes
end proc:
seq(A342420(n),n=1..20) ; # R. J. Mathar, Apr 06 2022

Table[PrimeNu[DivisorSigma[n, n]], {n, 30}]

a(n) = omega(sigma(n, n)); \\ Daniel Suteu, Mar 23 2022

from sympy import primefactors, factorint
def A352420(n): return len(set().union(*(primefactors((p**((e+1)*n)-1)//(p**n-1)) for p, e in factorint(n).items()))) # Chai Wah Wu, Mar 24 2022

a(n) = omega(sigma_n(n)) = A001221(A023887(n)).

a(67)-a(75) from Daniel Suteu, Mar 23 2022

A352839 Expansion of g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k).

Original entry on oeis.org

1, 1, 6, 39, 370, 4132, 59288, 990705, 19577018, 439550259, 11142216938, 313147651821, 9680830606850, 325944181383936, 11875777329091878, 465292113335910106, 19507503314546762246, 871248546067010133794, 41295079536653463057146

Offset: 0

Seiichi Manyama, Apr 05 2022

nonn

Cf. A129921, A180305, A320649.

Cf. A023887, A352841, A352842.

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

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

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

cp's OEIS Frontend

A157255 a(n) = sigma_(n^n)(n).

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A226890 E.g.f.: exp( Sum_{n>=1} sigma(n,n) * x^(n^2) / n^n ).

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PARI

Formula

A308763 a(n) = Sum_{d|n} d^(n-2).

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Mathematica

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PARI

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A319278 Square array sigma_k(n) read down antidiagonals: sum of the k-th powers of the divisors of n.

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Mathematica

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A321190 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k/(1 - x^k)).

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Maple

Mathematica

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A321295 a(n) = n * sigma_n(n).

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Magma

Mathematica

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Perl

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A346773 a(n) = Sum_{d|n} möbius(d)^n.

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PARI

PARI

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A350503 Self-convolution of sigma_n(n).

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