A034488 -id:A034488 - 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

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

Original entry on oeis.org

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

A241029 Sum of n-th powers of divisors of 22.

Original entry on oeis.org

4, 36, 610, 11988, 248914, 5314716, 115151530, 2513845188, 55090232674, 1209627165996, 26585860217050, 584603613083988, 12858141059430034, 282844580595234876, 6222201023261420170, 136884245263581500388, 3011407446068928780994

Offset: 0

Vincenzo Librandi, Apr 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (36,-343,792,-484).

Cf. sum of n-th powers of divisors of even k: A000051 (k=2), A001576 (k=4), A034488 (k=6), A034496 (k=8), A034517 (k=10), A034660 (k=12), A141013 (k=14), A020514 (k=16), A034661 (k=18), A034662 (k=20), this sequence (k=22), A034664 (k=24), A241030 (k=26), A241031 (k=28), A241032 (k=30), A034665 (k=32), A034666 (k=36), A034667 (k=40), A034668 (k=48), A034669 (k=56), A020516 (k=64), A034671 (k=72), A034672 (k=96), A034673 (k=120), A034674 (k=128), A034675 (k=144).

Magma
```
[DivisorSigma(n, 22): n in [0..20]];
```

Total[#^Range[0, 20]&/@Divisors[22]]
Table[(1 + 2^n) (1 + 11^n), {n, 0, 20}] (* Bruno Berselli, Apr 17 2014 *)
LinearRecurrence[{36,-343,792,-484},{4,36,610,11988},30] (* Harvey P. Dale, May 21 2014 *)

makelist((1+2^n)*(1+11^n), n, 0, 20); /* Bruno Berselli, Apr 17 2014 */

G.f.: 2*(2 - 54*x + 343*x^2 - 396*x^3)/((1 - x)*(1 - 2*x)*(1 - 11*x)*(1 - 22*x)). [Bruno Berselli, Apr 17 2014]

a(n) = (1 + 2^n)*(1 + 11^n). [Bruno Berselli, Apr 17 2014]

A224384 a(n) = 1 + 17^n.

Original entry on oeis.org

2, 18, 290, 4914, 83522, 1419858, 24137570, 410338674, 6975757442, 118587876498, 2015993900450, 34271896307634, 582622237229762, 9904578032905938, 168377826559400930, 2862423051509815794, 48661191875666868482, 827240261886336764178, 14063084452067724991010

Offset: 0

Philippe Deléham, Apr 05 2013

Sum of n-th powers of divisors of 17.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-17).

Cf. A001026.

17^Range[0,20]+1 (* Harvey P. Dale, Jul 19 2019 *)

a(n) = A001026(n) + 1.
G.f.: 1/(1-x) + 1/(1-17*x).
E.g.f.: exp(x) + exp(17*x).
a(n) = 18*a(n-1) - 17*a(n-2) with a(0) = 2, a(1) = 18.

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.

Showing 1-5 of 5 results.

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

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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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A241029 Sum of n-th powers of divisors of 22.

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A224384 a(n) = 1 + 17^n.

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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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