A034524 -id:A034524 - 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

A178248 a(n) = 12^n + 1.

Original entry on oeis.org

2, 13, 145, 1729, 20737, 248833, 2985985, 35831809, 429981697, 5159780353, 61917364225, 743008370689, 8916100448257, 106993205379073, 1283918464548865, 15407021574586369, 184884258895036417, 2218611106740436993

Offset: 0

Stuart Clary, Dec 20 2010

Prime factors of a(n) are in the Cunningham Project.

			a(3) = 12^3 + 1 = 1729.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. S. Wagstaff, Jr., The Cunningham Project.
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A000051, A001021, A007689, A016125, A024140, A034472, A034474, A034491, A034524, A052539, A062394, A062395, A062396, A062397, A063376, A063481, A074600-A074624, A228081.

[12^n + 1: n in [0..20]]; // Vincenzo Librandi, May 02 2011

nmax = 17; 12^Range[0, nmax] + 1
LinearRecurrence[{13,-12},{2,13},20] (* Harvey P. Dale, Nov 02 2015 *)

vector(21,n,12^(n-1)+1) \\ Charles R Greathouse IV, May 02 2011

O.g.f.: (2-13*x)/(1-13*x+12*x^2) = (2-13*x)/((1-x)(1-12*x)).
E.g.f.: exp(x) + exp(12*x). - Stefano Spezia, Mar 20 2023
From Elmo R. Oliveira, Dec 15 2023: (Start)
a(n) = 12*a(n-1) - 11 for n>0.
a(n) = 13*a(n-1) - 12*a(n-2) for n>1.
a(n) = A001021(n)+1 = A024140(n)+2.
a(n) = (11*A016125(n) + 13)/12. (End)

A366690 a(n) = phi(11^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 60, 432, 7320, 53680, 803520, 6495720, 100874752, 764738496, 12756110400, 89493288192, 1568774615040, 11278053084480, 180228847518720, 1310982643872000, 22974417331646464, 168479281019744640, 2521788545778163200, 20190830281379049600

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A000010, A062308, A366685, A366686, A366687, A366689, A366689, A366716.

EulerPhi[11^Range[0,19] + 1] (* Paul F. Marrero Romero, Nov 10 2023 *)

PARI
```
{a(n) = eulerphi(11^n+1)}
```

a(n) = A000010(A034524(n)). - Paul F. Marrero Romero, Nov 10 2023

A366686 Number of distinct prime divisors of 11^n + 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 3, 6, 4, 5, 5, 6, 3, 5, 5, 6, 4, 5, 4, 6, 7, 5, 3, 6, 6, 5, 6, 6, 4, 11, 6, 9, 7, 4, 4, 9, 5, 5, 9, 4, 6, 10, 6, 6, 5, 7, 6, 9, 3, 6, 9, 12, 7, 10, 6, 6, 8, 5, 4, 10, 3, 9, 8, 8, 7, 12, 8, 5, 10, 7, 8, 11, 6, 11, 11, 6, 10, 9, 5

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A001221, A366681, A102347, A366687, A366688, A366689, A366690.

Cf. A046799, A119704, A366712.

for(n = 0, 100, print1(omega(11^n + 1), ", "))

a(n) = omega(11^n+1) = A001221(A034524(n)).

A366689 Sum of the divisors of 11^n+1.

Original entry on oeis.org

3, 28, 186, 3458, 21966, 375816, 2911272, 45470096, 340452396, 6278429920, 39543942612, 706019328000, 4708961513592, 82162955169792, 599236951715280, 11195197038864384, 68925937595777100, 1179397832668228992, 9136813499663186064, 144079834776308121600

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

			a(4)=21966 because 11^4+1 has divisors {1, 2, 7321, 14642}.

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A000203, A366666, A366684, A366686, A366687, A366688, A366690, A366715.

a:=n->numtheory[sigma](11^n+1):
seq(a(n), n=0..100);

DivisorSigma[1,11^Range[0,20]+1] (* Harvey P. Dale, Jun 22 2025 *)

a(n) = sigma(11^n+1) = A000203(A034524(n)).

A366688 Number of divisors of 11^n+1.

Original entry on oeis.org

2, 6, 4, 18, 4, 12, 16, 12, 8, 48, 8, 96, 16, 48, 32, 144, 8, 48, 32, 96, 16, 72, 16, 96, 128, 48, 8, 240, 64, 48, 64, 96, 16, 4608, 64, 1152, 128, 24, 16, 1152, 32, 48, 512, 24, 64, 3072, 64, 96, 32, 192, 64, 1152, 8, 96, 512, 6144, 128, 2304, 64, 96, 256, 48

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

			a(4)=4 because 11^4+1 has divisors {1, 2, 7321, 14642}.

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A000005, A062308, A366683, A366686, A366687, A366689, A366690.

Cf. A046798, A344897, A366714.

a:=n->numtheory[tau](11^n+1):
seq(a(n), n=0..100);

DivisorSigma[0,11^Range[0,70]+1] (* Harvey P. Dale, Mar 17 2025 *)

PARI
```
a(n) = numdiv(11^n+1);
```

a(n) = sigma0(11^n+1) = A000005(A034524(n)).

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

A366687 Number of prime factors of 11^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 3, 2, 5, 2, 4, 4, 4, 3, 7, 3, 7, 4, 6, 5, 8, 3, 6, 5, 7, 4, 7, 4, 7, 7, 6, 3, 10, 6, 6, 6, 7, 4, 13, 6, 11, 7, 5, 4, 11, 5, 6, 9, 5, 6, 13, 6, 7, 5, 8, 6, 11, 3, 7, 9, 13, 7, 12, 6, 7, 8, 6, 4, 13, 3, 10, 8, 9, 7, 14, 8, 6, 10, 8, 8, 13, 6, 12, 12, 7, 10

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A001222, A057934, A062308, A366686, A366688, A366689, A366690.

Cf. A054992, A057941, A057940, A057939, A057938, A057937, A057936, A057935, A057934, A366713.

Mathematica
```
PrimeOmega[11^Range[70]+1]
```
PARI
```
a(n)=bigomega(11^n+1)
```

a(n) = bigomega(11^n+1) = A001222(A034524(n)).

A130652 a(n) = 11^n - 2.

Original entry on oeis.org

9, 119, 1329, 14639, 161049, 1771559, 19487169, 214358879, 2357947689, 25937424599, 285311670609, 3138428376719, 34522712143929, 379749833583239, 4177248169415649, 45949729863572159, 505447028499293769, 5559917313492231479, 61159090448414546289, 672749994932560009199

Offset: 1

Alexander Adamchuk, Jun 20 2007

There are only two known primes in a(n): a(4) = 14639 and a(6) = 1771559 (see A128472 = smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists). 3 divides a(2k-1). 7 divides a(3k-1). 13 divides a(12k-5). 17 divides a(16k-14).
Final digit of a(n) is 9.
Final two digits of a(n) are periodic with period 10: a(n) mod 100 = {09, 19, 29, 39, 49, 59, 69, 79, 89, 99}.
Final three digits of a(n) are periodic with period 50: a(n) mod 1000 = {009, 119, 329, 639, 049, 559, 169, 879, 689, 599, 609, 719, 929, 239, 649, 159, 769, 479, 289, 199, 209, 319, 529, 839, 249, 759, 369, 079, 889, 799, 809, 919, 129, 439, 849, 359, 969, 679, 489, 399, 409, 519, 729, 039, 449, 959, 569, 279, 089, 999}.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A001020, A024127, A034524. Cf. A104096 = Largest prime <= 11^n. Cf. A084714 = smallest prime of the form (2n-1)^k - 2, or 0 if no such number exists. Cf. A128472 = smallest prime of the form (2n-1)^k - 2 for k>(2n-1), or 0 if no such number exists. Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.

[11^n - 2: n in [1..50]]; // Vincenzo Librandi, Jun 08 2011

LinearRecurrence[{12, -11},{9, 119},17] (* Ray Chandler, Aug 26 2015 *)

a(n) = 11*a(n-1) + 20; a(1)=9. - Vincenzo Librandi, Jun 08 2011
From Elmo R. Oliveira, Jun 16 2025: (Start)
G.f.: x*(11*x+9)/((11*x-1)*(x-1)).
E.g.f.: 1 + exp(x)*(exp(10*x) - 2).
a(n) = 12*a(n-1) - 11*a(n-2) for n > 2. (End)

A034659 a(n) = (11^n + 1)/2.

Original entry on oeis.org

1, 6, 61, 666, 7321, 80526, 885781, 9743586, 107179441, 1178973846, 12968712301, 142655835306, 1569214188361, 17261356071966, 189874916791621, 2088624084707826, 22974864931786081, 252723514249646886, 2779958656746115741, 30579545224207273146, 336374997466280004601

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A034524.

a(n)=(11^n+1)/2 \\ Charles R Greathouse IV, Aug 23 2024

a(n) = 11*a(n-1) - 5 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
G.f.: (1-6*x)/((1-x)*(1-11*x)). - Colin Barker, May 04 2012
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(6*x)*cosh(5*x).
a(n) = A034524(n)/2.
a(n) = 12*a(n-1) - 11*a(n-2) for n > 1. (End)

a(18)-a(20) from Elmo R. Oliveira, Aug 23 2024

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

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A178248 a(n) = 12^n + 1.

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A366690 a(n) = phi(11^n+1), where phi is Euler's totient function (A000010).

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PARI

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A366686 Number of distinct prime divisors of 11^n + 1.

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PARI

Formula

A366689 Sum of the divisors of 11^n+1.

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Maple

Mathematica

Formula

A366688 Number of divisors of 11^n+1.

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Programs

Maple

Mathematica

PARI

Formula

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

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Maple