A023887 -id:A023887 - cp's OEIS frontend

A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

1, 2, 3, 2, 4, 10, 3, 7, 21, 73, 2, 6, 26, 126, 626, 4, 12, 50, 252, 1394, 8052, 2, 8, 50, 344, 2402, 16808, 117650, 4, 15, 85, 585, 4369, 33825, 266305, 2113665, 3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 4, 18, 130, 1134, 10642, 103158, 1015690, 10078254, 100390882, 1001953638

Offset: 1

Reinhard Zumkeller, May 21 2003

			From _R. J. Mathar_, Dec 06 2006 (Start):
The triangle may be extended to a rectangular array (A319278):
  1  1   1    1     1 1 1 1 1 1 1 ...
  2  3   5    9    17 33 65 129 257 513 1025 ...
  2  4  10   28    82 244 730 2188 6562 19684 59050 ...
  3  7  21   73   273 1057 4161 16513 65793 262657 1049601 ...
  2  6  26  126   626 3126 15626 78126 390626 1953126 9765626 ...
  4 12  50  252  1394 8052 47450 282252 1686434 10097892 60526250 ...
  2  8  50  344  2402 16808 117650 823544 5764802 40353608 282475250 ...
  4 15  85  585  4369 33825 266305 2113665 16843009 134480385 1074791425 ...
  3 13  91  757  6643 59293 532171 4785157 43053283 387440173 3486843451 ...
  4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638... (End)

T. D. Noe, Rows n=1..100, flattened
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963.
Cf. A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.

T:= (n,k)-> numtheory[sigma][k](n):
seq(seq(T(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Oct 25 2024

T[n_, k_] := DivisorSigma[k, n];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

row(n) = {my(f = factor(n)); vector(n, k, sigma(f, k-1));} \\ Amiram Eldar, May 09 2025

t(n, k) = Product(((p^((e(n, p)+1)*k))-1)/(p^k-1): n=Product(p^e(n, p): p prime)), 0<=k

t(n,0) = A000005(n), t(n,n) = A023887(n).

t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.

t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.

From R. J. Mathar, Oct 29 2006: (Start)

t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);

t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);

t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);

t(11,k) = A034524(k); t(12,k) = A034660(k). (End)

Corrected by R. J. Mathar, Dec 05 2006

A245466 a(n) = sigma_1(1) + sigma_2(2) + sigma_3(3) + ... + sigma_n-1(n-1) + sigma_n(n).

Original entry on oeis.org

1, 6, 34, 307, 3433, 50883, 874427, 17717436, 405157609, 10414924259, 295726594871, 9214021138217, 312089127730471, 11424774176377721, 449318695089164129, 18896344248070459234, 846136606134407223412, 40192694877626586149007, 2018612350537940175272987

Offset: 1

Wesley Ivan Hurt, Jul 22 2014

nonn

Let sigma_k(n) represent the sum of the k-th powers of the divisors of n.
Then a(n) = Sum_{k=1..n} sigma_k(k), the partial sums of sigma_k(k) for k from 1 to n.
Partial sums of A023887.

			a(1) = 1 because sigma_1(1) = sigma(1) = 1.
a(2) = 6: sigma_1(1) + sigma_2(2) = 1 + (1^2 + 2^2) = 6.
a(3) = 34: sigma_1(1) + sigma_2(2) + sigma_3(3) = 6 + (1^3 + 3^3) = 34.
a(4) = 307: sigma_1(1) + ... + sigma_4(4) = 34 + (1^4 + 2^4 + 4^4) = 307.

Jens Kruse Andersen, Table of n, a(n) for n = 1..100

Cf. A000203, A001157-A001160, A013954-A013958, A023887.

[&+[DivisorSigma(i, i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Jul 29 2014

[n eq 1 select 1 else Self(n-1)+ DivisorSigma(n, n): n in [1..20]]; // Vincenzo Librandi, Aug 05 2015

B:= [seq(numtheory:-sigma[n](n),n=1..100)]:
seq(add(B[i],i=1..n),n=1..100); # Robert Israel, Jul 28 2014

Table[Sum[DivisorSigma[k, k], {k, n}], {n, 20}]
Accumulate[Table[DivisorSigma[n,n],{n,20}]] (* Harvey P. Dale, Apr 10 2018 *)

a(n) = sum(i=1,n,sigma(i,i))
vector(50, n, a(n)) \\ Derek Orr, Jul 27 2014

a(n) = Sum_{k=1..n} sigma_k(k).
a(1) = 1. a(n) = a(n-1) + sigma_n(n), for n > 1. - Jens Kruse Andersen, Jul 29 2014
a(n) = n + Sum_{d=2..n} (d^(d*(floor(n/d)+1))-d^d)/(d^d-1). - Chayim Lowen, Aug 04 2015

A363646 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^2 - 1).

Original entry on oeis.org

2, 11, 58, 565, 6256, 95762, 1647094, 33752329, 774919720, 20029303030, 570623341234, 17838801038274, 605750213184520, 22226048320465666, 875787902918124708, 36894332593824661521, 1654480523772673528372, 78693266840741507386757

Offset: 1

Seiichi Manyama, Jun 13 2023

nonn

Cf. A023887, A338663, A363647, A363648.

Cf. A007503, A362683.

a[n_] := DivisorSum[n, (n/#)^n * (# + 1) &]; Array[a, 20] (* Amiram Eldar, Jul 17 2023 *)

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

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

A292919 Sum of n-th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 28, 1, 3126, 730, 823544, 1, 387440173, 9765626, 285311670612, 531442, 302875106592254, 678223072850, 437893920912786408, 1, 827240261886336764178, 150094635684419611, 1978419655660313589123980, 95367431640626, 5842587018944528395924761632, 81402749386839761113322

Offset: 1

Ilya Gutkovskiy, Sep 26 2017

Seiichi Manyama, Table of n, a(n) for n = 1..386
Index entries for sequences related to sums of divisors

Diagonal of A285425.

Cf. A000593, A023887, A082245.

f:= proc(n) local t,d;
  t:= n/2^padic:-ordp(n,2);
  add(d^n, d = numtheory:-divisors(t));
end proc:
map(f, [$1..30]); # Robert Israel, Sep 27 2017

Rest[Table[SeriesCoefficient[Sum[(2 k - 1)^n x^(2 k - 1)/(1 - x^(2 k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 22}]]
f[n_] := Plus @@ (Select[Divisors[n], OddQ]^n); Array[f, 22] (* Robert G. Wilson v, Sep 26 2017 *)

a(n) = sumdiv(n, d, if (d%2, d^n)); \\ Michel Marcus, Sep 08 2018

a(n) = [x^n] Sum_{k>=1} (2*k - 1)^n*x^(2*k-1)/(1 - x^(2*k-1)).

a(2^k) = 1.

A308504 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).

Original entry on oeis.org

1, 1, 5, 1, 9, 28, 1, 17, 82, 273, 1, 33, 244, 1057, 3126, 1, 65, 730, 4161, 15626, 47450, 1, 129, 2188, 16513, 78126, 282252, 823544, 1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173

Offset: 1

Seiichi Manyama, Jun 02 2019

			a(4) = a(2*3/2 + 1) = sigma_3(1) = 1.
a(5) = a(2*3/2 + 2) = sigma_3(2) = 1^3 + 2^3 = 9.
a(6) = a(2*3/2 + 3) = sigma_3(3) = 1^3 + 3^3 = 28.
Square array begins:
       1,      1,       1,        1,        1, ...
       5,      9,      17,       33,       65, ...
      28,     82,     244,      730,     2188, ...
     273,   1057,    4161,    16513,    65793, ...
    3126,  15626,   78126,   390626,  1953126, ...
   47450, 282252, 1686434, 10097892, 60526250, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Index entries for sequences related to sigma(n)

Columns k=0..2 give A023887, A294645, A294810.
A(n,n) gives A308570.
Cf. A279394, A308502.

T[n_, k_] := DivisorSum[n, #^(n+k) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

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

a((i-1)*i/2 + j) = sigma_i(j) for 1 <= j <= i.

A338685 a(n) = Sum_{d|n} d^n * binomial(d, n/d).

Original entry on oeis.org

1, 8, 81, 1040, 15625, 282123, 5764801, 134610944, 3486804084, 100097656250, 3138428376721, 107025924222976, 3937376385699289, 155582338242342053, 6568408660888671875, 295155786482995691520, 14063084452067724991009, 708240750793407501694308, 37589973457545958193355601

Offset: 1

Seiichi Manyama, Apr 23 2021

nonn

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

Cf. A023887, A318636, A327238, A338684, A338693.

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

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

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

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

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

A363648 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^4 - 1).

Original entry on oeis.org

4, 26, 128, 1219, 12556, 195278, 3294292, 67773349, 1550075836, 40097713880, 1141246682808, 35686524105658, 1211500426369572, 44454809534927314, 1751576172678539608, 73789791194939982793, 3308961047545347057848, 157387135278770854655312

Offset: 1

Seiichi Manyama, Jun 13 2023

nonn

Cf. A023887, A338663, A363646, A363647.

Cf. A363640.

a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + 3, 3] &]; Array[a, 20] (* Amiram Eldar, Jul 17 2023 *)

a(n) = sumdiv(n, d, (n/d)^n*binomial(d+3, 3));

a(n) = Sum_{d|n} (n/d)^n * binomial(d+3,3).

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.

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A342629 a(n) = Sum_{d|n} (n/d)^(n-d).

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A363647 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^3 - 1).

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A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

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A245466 a(n) = sigma_1(1) + sigma_2(2) + sigma_3(3) + ... + sigma_n-1(n-1) + sigma_n(n).

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A363646 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^2 - 1).

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A292919 Sum of n-th powers of odd divisors of n.

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A308504 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).

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