A108639 -id:A108639 - 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

A279394 Triangle read by rows, T(n,m) = sigma_{n-m}(m) for n >= 1, m = 1,2, ..., n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 7, 2, 1, 17, 28, 21, 6, 4, 1, 33, 82, 73, 26, 12, 2, 1, 65, 244, 273, 126, 50, 8, 4, 1, 129, 730, 1057, 626, 252, 50, 15, 3, 1, 257, 2188, 4161, 3126, 1394, 344, 85, 13, 4, 1, 513, 6562, 16513, 15626, 8052, 2402, 585, 91, 18, 2, 1, 1025, 19684, 65793, 78126, 47450, 16808, 4369, 757, 130, 12, 6

Offset: 1

Wolfdieter Lang, Jan 07 2017

See A109974 (downward antidiagonals) for details and references. sigma_k(n) is the sum of the k-th power of the positive divisors of n.
This is the triangle read by rows obtained from the array sigma_k(n) for k >= 0, n >= 1, read by upward antidiagonals.
The row sums are A108639.

			The triangle T(n, m) begins:
n\m 1   2    3    4    5    6   7  8  9 10
1:  1
2:  1   2
3:  1   3    2
4:  1   5    4    3
5:  1   9   10    7    2
6:  1  17   28   21    6    4
7:  1  33   82   73   26   12   2
8:  1  65  244  273  126   50   8  4
9:  1 129  730 1057  626  252  50 15  3
10: 1 257 2188 4161 3126 1394 344 85 13  4
...
n = 11: 1 513 6562 16513 15626 8052 2402 585 91 18 2,
n = 12: 1 1025 19684 65793 78126 47450 16808 4369 757 130 12 6.
...

Cf. A109974, A108639.

T := (n, k) -> numtheory:-sigma[n-k](k):
seq(seq(T(n,k), k=1..n), n=1..12); # Peter Luschny, Jan 07 2017

Table[DivisorSigma[k, #] &[n - k + 1], {n, 0, 11}, {k, n, 0, -1}] (* Michael De Vlieger, Jan 09 2017 *)

T(n, m) = sigma_{n-m}(m), n >= 1, m = 1..n.

A108640 a(n) = Product_{k=1..n} sigma_{n-k}(k), where sigma_m(k) = sum{j|k} j^m.

Original entry on oeis.org

1, 2, 6, 60, 1260, 239904, 123263712, 872883648000, 35330106763980000, 15502816844111220549120, 32196148399600498119169883520, 2560463149313858442381787649990400000, 717635502576022020068175045395317927056000000

Offset: 1

Leroy Quet, Jul 06 2005

nonn

			a(5) = 1^4 * (1^3 +2^3) * (1^2 +3^2) * (1^1 +2^1 +4^1) * (1^0 +5^0) = 1 * 9 * 10 * 7 * 2 = 1260.

G. C. Greubel, Table of n, a(n) for n = 1..44

Cf. A108639 (with sums).

A108639:= func< n | (&*[DivisorSigma(j, n-j): j in [0..n-1]]) >;
[A108639(n): n in [1..30]]; // G. C. Greubel, Oct 18 2023

with(numtheory): s:=proc(n,k) local div: div:=divisors(n): sum(div[j]^k,j=1..tau(n)) end: a:=n->product(s(i,n-i),i=1..n): seq(a(n),n=1..14); # Emeric Deutsch, Jul 13 2005

Table[Product[DivisorSigma[j,n-j], {j,0,n-1}], {n,30}] (* G. C. Greubel, Oct 18 2023 *)

a(n) = prod(k=1, n, sigma(k, n-k)); \\ Michel Marcus, Aug 16 2019

def A108640(n): return product(sigma(n-j,j) for j in range(n))
[A108640(n) for n in range(1,31)] # G. C. Greubel, Oct 18 2023

More terms from Emeric Deutsch, Jul 13 2005

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A279394 Triangle read by rows, T(n,m) = sigma_{n-m}(m) for n >= 1, m = 1,2, ..., n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A108640 a(n) = Product_{k=1..n} sigma_{n-k}(k), where sigma_m(k) = sum{j|k} j^m.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Extensions