A082245 -id:A082245 - cp's OEIS frontend

A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.

1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 2, 7, 10, 9, 1, 4, 6, 21, 28, 17, 1, 2, 12, 26, 73, 82, 33, 1, 4, 8, 50, 126, 273, 244, 65, 1, 3, 15, 50, 252, 626, 1057, 730, 129, 1, 4, 13, 85, 344, 1394, 3126, 4161, 2188, 257, 1, 2, 18, 91, 585, 2402, 8052, 15626, 16513, 6562, 513, 1

Offset: 0

Paul Barry, Jul 06 2005

Rows sums are A108639. Antidiagonal sums are A109976. Matrix inverse is A109977.
From Wolfdieter Lang, Jan 29 2016: (Start)
The sum of the (k-1)th power of the divisors of n, sigma_(k-1)(n), appears also as eigenvalue lambda(k, n) of the Hecke operators T_n, n a positive integer, acting on the normalized Eisenstein series E_k(q) = ((2*Pi*i)^k/((k-1)!*Zeta(k))*G_k(q) with even k >= 4 and q = 2*Pi*i*z, where z is from the upper half of the complex plane: T_n E_k = sigma_(k-1)(n)*E_k. These Eisenstein series are entire modular forms of weight k, and each E_k(q) is a simultaneous eigenform of the Hecke operators T_n, for every n >= 1.
This results from the Fourier coefficients of E_k(q) = Sum_{m>=0} E(k, m)*q^m, with E(k, 0) =1 and E(k, m) = ((2*Pi*i)^k / ((k-1)!*Zeta(k))* sigma_(k-1)(m) for m >= 1, together with the Fourier coefficients of T_n E_k. The eigenvalues lambda(n, k) = (Sum_{d | gcd(n,m)} d^{k-1}*E(k, m*n/d^2)) / E(k, m) for each m >= 0. For m=0 this becomes lambda(n, k) = sigma_(k-1)(n).
For Hecke operators, Fourier coefficients and simultaneous eigenforms see, e.g., the Koecher - Krieg reference, p. 207, eqs. (5) and (6) and p. 211, section 4, or the Apostol reference, p. 120, eq. (13), pp. 129 - 134. (End)

			Start of array:
  1,  2,  2,   3,   2,    4, ...
  1,  3,  4,   7,   6,   12, ...
  1,  5, 10,  21,  26,   50, ...
  1,  9, 28,  73, 126,  252, ...
  1, 17, 82, 273, 626, 1394, ...
  ...
The triangle T(m, k) with row offset 1 starts:
  m\k 0  1  2   3    4    5    6    7   8  9 ...
  1:  1
  2:  2  1
  3:  2  3  1
  4:  3  4  5   1
  5:  2  7 10   9    1
  6:  4  6 21  28   17    1
  7:  2 12 26  73   82   33    1
  8:  4  8 50 126  273  244   65    1
  9:  3 15 50 252  626 1057  730  129   1
  10: 4 13 85 344 1394 3126 4161 2188 257  1
  ... - _Wolfdieter Lang_, Jan 14 2016

Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 120, 129 - 134.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.

Alois P. Heinz, Antidiagonals k = 0..140, flattened

Rows: A000005, A000203, A001157, A001158, A001159, A001160, A013954 - A013972.
Columns: A000051, A034472, A001576, A034474, A034488, A034491, A034496, A034513, A034517, A034524, A034660.
Row sums A108639.
Diagonals A082245, A023887.
Related sequences: A082771, A109976, A109977, A109978.

A109974:= func< n,k | DivisorSigma(k-1, n-k+1) >;
[A109974(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 18 2023

with(numtheory):
seq(seq(sigma[k](1+d-k), k=0..d), d=0..12);  # Alois P. Heinz, Feb 06 2013

rows=12; Flatten[Table[DivisorSigma[k-n, n], {k,1,rows}, {n,k,1,-1}]] (* Jean-François Alcover, Nov 15 2011 *)

def A109974(n,k): return sigma(n-k+1, k-1)
flatten([[A109974(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 18 2023

Regarded as a triangle, T(n, k) = if(k<=n, sigma(k-1, n-k+1), 0). - Franklin T. Adams-Watters, Jul 17 2006
If the row index (the index of the antidiagonal of the array) is taken as m with offset 1 the triangle is T(m, k) = sigma_k(m-k), 1 <= k+1 <= m, otherwise 0. - Wolfdieter Lang, Jan 14 2016
G.f. for the triangle with offset 1: G(x,y) = Sum_{j>=1} x^j/((1-x^j)*(1-j*x*y)). - Robert Israel, Jan 14 2016

A294645 a(n) = Sum_{d|n} d^(n+1).

Original entry on oeis.org

1, 9, 82, 1057, 15626, 282252, 5764802, 134480385, 3486843451, 100048830174, 3138428376722, 107006334784468, 3937376385699290, 155572843119354936, 6568408508343827972, 295150156996346511361, 14063084452067724991010, 708236696816416252145973

Offset: 1

Seiichi Manyama, Nov 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Divisor Function

Column k=1 of A308504.

Cf. A023882, A023887, A082245, A158095, A292312.

Table[DivisorSigma[n + 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Oct 07 2020 *)

PARI
```
{a(n) = sigma(n, n+1)}
```

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

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-(k*x)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>0} k^(k+1)*x^k/(1-(k*x)^k).
L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 02 2019
a(n) ~ n^(n+1). - Vaclav Kotesovec, Oct 07 2020

A359112 a(n) = Sum_{d|n} (n/d) * d^(n-d).

Original entry on oeis.org

1, 3, 4, 13, 6, 109, 8, 777, 2197, 7541, 12, 374809, 14, 1675773, 31954096, 100794385, 18, 7391871271, 20, 163547770441, 2037381161992, 570634875581, 24, 1275177760626097, 476837158203151, 605750431288341, 450286447756825720, 2258377795760750777, 30

Offset: 1

Seiichi Manyama, Dec 17 2022

nonn

Cf. A000203, A167531.

Cf. A082245, A342628.

a[n_] := DivisorSum[n, #^(n-#)*n/# &]; Array[a, 29] (* Amiram Eldar, Aug 27 2023 *)

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

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)^2))

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

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

A308814 a(n) = Sum_{d|n} n^(d-1).

Original entry on oeis.org

1, 3, 10, 69, 626, 7819, 117650, 2097673, 43046803, 1000010011, 25937424602, 743008621405, 23298085122482, 793714780783695, 29192926025441476, 1152921504875286545, 48661191875666868482, 2185911559749718382455, 104127350297911241532842, 5242880000000512000168021

Offset: 1

Seiichi Manyama, Jun 26 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A066108, A082245, A262843, A308813.

Table[Total[n^(Divisors[n]-1)],{n,20}] (* Harvey P. Dale, Aug 08 2019 *)

PARI
```
{a(n) = sumdiv(n, d, n^(d-1))}
```

a(n) = A308813(n,n).
a(n) = A066108(n)/n.
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 05 2021

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.

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).

A158265 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n+1)x^n/n ).

Original entry on oeis.org

1, 2, 11, 74, 697, 8002, 115158, 1949640, 38662510, 872245634, 22150393253, 623661939852, 19296665400632, 650198159192554, 23700604926216759, 928939297013294294, 38956230043045053042, 1740248411222193973416

Offset: 0

Paul D. Hanna, Mar 29 2009

nonn

Definition: sigma(n,n+1) = Sum_{d|n} d^(n+1): [1,9,82,1057,15626,...].

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 74*x^3 + 697*x^4 + 8002*x^5 +...
log(A(x)) = 2*x + 18*x^2/2 + 164*x^3/3 + 2114*x^4/4 + 31252*x^5/5 +...

Cf. A158095, A023881, A082245.

a(n)=polcoeff(exp(sum(m=1, n, 2*sigma(m, m+1)*x^m/m)+x*O(x^n)), n)

a(n) ~ 2 * exp(1) * n^(n-1). - Vaclav Kotesovec, Oct 31 2024

A359701 a(n) = Sum_{d|n} d^(d + n/d - 2).

Original entry on oeis.org

1, 3, 10, 69, 626, 7812, 117650, 2097425, 43046803, 1000003158, 25937424602, 743008418676, 23298085122482, 793714774077816, 29192926025406980, 1152921504623628545, 48661191875666868482, 2185911559739084235093, 104127350297911241532842

Offset: 1

Seiichi Manyama, Jan 11 2023

nonn

Cf. A082245, A124923, A262843, A294956.

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

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

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

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

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

cp's OEIS Frontend

A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.

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A294645 a(n) = Sum_{d|n} d^(n+1).

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A359112 a(n) = Sum_{d|n} (n/d) * d^(n-d).

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A308814 a(n) = Sum_{d|n} n^(d-1).

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

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Maple

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

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A158265 G.f.: A(x) = exp( Sum_{n>=1} 2*sigma(n,n+1)*x^n/n ).

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A158265 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n+1)x^n/n ).