A024140 -id:A024140 - cp's OEIS frontend

A001021 Powers of 12.

1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, 1283918464548864, 15407021574586368, 184884258895036416, 2218611106740436992

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 12), L(1, 12), P(1, 12), T(1, 12). Essentially same as Pisot sequences E(12, 144), L(12, 144), P(12, 144), T(12, 144). See A008776 for definitions of Pisot sequences.
Central terms of the triangle in A100851. - Reinhard Zumkeller, Mar 04 2006
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 12-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Starting with 12, this sequence appears in the film "Vollmond" (1998, dir. Fredi Murer), when the children write it on the sidewalk at night. - Alonso del Arte, Dec 21 2011

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 276.
Tanya Khovanova, Recursive Sequences.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A024140 and A178248, 12^n +/- 1; A016125.

Cf. A000005, A000244, A000302, A000351, A000384, A001222, A008585, A008776, A100851, A159991.

a001021 = (12 ^)
a001021_list = iterate (* 12) 1  -- Reinhard Zumkeller, Mar 31 2012

A001021:=-1/(-1+12*z); # Simon Plouffe in his 1992 dissertation

Table[12^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=12^n \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 1/(1-12*x).
E.g.f.: exp(12x).
a(n) = 12*a(n-1). - Zerinvary Lajos, Apr 27 2009
a(n) = A159991(n)/A000351(n). - Reinhard Zumkeller, May 02 2009
From Reinhard Zumkeller, Mar 31 2012: (Start)
a(n) = A000302(n) * A000244(n). - Reinhard Zumkeller, Mar 31 2012
A001222(a(n)) = A008585(n); A000005(a(n)) = A000384(a(n)). (End)
a(n) = det(|ps(i+2, j)|, 1 <= i, j <= n), where ps(n, k) are Legendre-Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013

A016125 Expansion of 1/((1-x)(1-12x)).

Original entry on oeis.org

1, 13, 157, 1885, 22621, 271453, 3257437, 39089245, 469070941, 5628851293, 67546215517, 810554586205, 9726655034461, 116719860413533, 1400638324962397, 16807659899548765, 201691918794585181

Offset: 0

N. J. A. Sloane

Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=12, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det(A). - Milan Janjic, Feb 21 2010
Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=13, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-2)=(-1)^n*charpoly(A,1). - Milan Janjic, Feb 21 2010
Numbers that are repunits in duodecimal representation. - Reinhard Zumkeller, Dec 12 2012
a(n) is the total number of holes in a certain box fractal (start with 12 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015

			For n=5, a(5) = 1*6 + 11*15 + 121*20 + 1331*15 + 14641*6 + 161051*1 = 271453. - _Bruno Berselli_, Nov 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Duodecimal
Wikipedia, Duodecimal
Wikipedia, Repunit
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A001021, A024140, A178248.

Cf. A001020, A135278.

a016125 n = a016125_list !! n
a016125_list = iterate ((+ 1) . (* 12)) 1
-- Reinhard Zumkeller, Dec 12 2012

[(12^(n+1)-1)/11: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011

a:=n->sum(12^(n-j),j=1..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=13},Table[c=13*b-12*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[1/((1-x)(1-12x)),{x,0,20}],x] (* or *) LinearRecurrence[{13,-12},{1,13},20] (* Harvey P. Dale, Aug 20 2022 *)

A016125(n):=(12^(n+1) - 1)/11$
makelist(A016125(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

Vec(1/(1-13*x+12*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011

[lucas_number1(n,13,12) for n in range(1,18)] # Zerinvary Lajos, Apr 29 2009

[gaussian_binomial(n,1,12) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

[(12^(n+1)-1)/11 for n in (0..20)] # Bruno Berselli, Nov 11 2015

a(n) = (12^(n+1) - 1)/11.
a(n) = 12*a(n-1)+1 for n>0, a(0)=1. - Vincenzo Librandi, Nov 19 2010
a(n) =  Sum_{i=0...n} 11^i*binomial(n+1,n-i). - Bruno Berselli, Nov 11 2015
E.g.f.: exp(x)*(12*exp(11*x) - 1)/11. - Stefano Spezia, Mar 11 2023

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)

A366709 Number of divisors of 12^n-1.

Original entry on oeis.org

2, 4, 4, 16, 4, 32, 8, 64, 16, 16, 12, 256, 8, 64, 64, 512, 8, 512, 4, 192, 32, 48, 16, 4096, 16, 192, 64, 1024, 32, 8192, 32, 2048, 192, 64, 512, 16384, 8, 64, 128, 12288, 16, 12288, 32, 3072, 4096, 256, 8, 262144, 32, 1024, 64, 6144, 128, 65536, 192, 8192

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

			a(3)=4 because 12^3-1 has divisors {1, 11, 157, 1727}.

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A000005, A024140, A046801, A070528, A366707, A366708, A366710, A366683.

a:=n->numtheory[tau](12^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 12^Range[100]-1]
```
PARI
```
a(n) = numdiv(12^n-1);
```

a(n) = sigma0(12^n-1) = A000005(A024140(n)).

A366707 Number of distinct prime divisors of 12^n - 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 3, 8, 3, 6, 6, 9, 3, 9, 2, 7, 5, 5, 4, 12, 4, 7, 6, 10, 5, 13, 5, 11, 7, 6, 9, 14, 3, 6, 7, 13, 4, 13, 5, 11, 12, 8, 3, 18, 5, 10, 6, 12, 7, 16, 7, 13, 7, 8, 4, 18, 4, 8, 8, 13, 8, 16, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 10, 17, 4

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A024140, A001221, A046800, A133801, A102347, A250288, A252170, A366708, A366709, A366681.

for(n = 1, 100, print1(omega(12^n - 1), ", "))

a(n) = omega(12^n-1) = A001221(A024140(n)).

A366710 Sum of the divisors of 12^n-1.

Original entry on oeis.org

12, 168, 1896, 30240, 271464, 4247040, 39156480, 636854400, 5817876000, 72749094432, 852203639280, 15743437516800, 116720110574544, 1518251476008960, 17220536137159296, 292933954031846400, 2420303924088730368, 38936041113123840000, 348523635677043192936

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

			a(3)=1896 because 12^3-1 has divisors {1, 11, 157, 1727}.

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A024140, A000203, A366707, A366708, A366709, A366711, A366684.

a:=n->numtheory[sigma](12^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 12^Range[30]-1]
```

a(n) = sigma(12^n-1) = A000203(A024140(n)).

A366711 a(n) = phi(12^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

10, 120, 1560, 13440, 226200, 2021760, 32518360, 274391040, 4534807680, 51953616000, 646094232960, 4662793175040, 97266341877120, 1070382142166400, 13666309113600000, 109897747141754880, 2016918439151095000, 17518491733377024000, 290436363064202660760

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), this sequence (k=12).

Cf. A000010, A024140, A366707, A366708, A366709, A366710, A366717, A366718.

Mathematica
```
EulerPhi[12^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(12^n-1)}
```

A366708 Number of prime factors of 12^n - 1 (counted with multiplicity).

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 3, 6, 4, 4, 4, 8, 3, 6, 6, 9, 3, 9, 2, 8, 5, 6, 4, 12, 4, 8, 6, 10, 5, 13, 5, 11, 8, 6, 9, 14, 3, 6, 7, 14, 4, 14, 5, 12, 12, 8, 3, 18, 5, 10, 6, 13, 7, 16, 8, 13, 7, 8, 4, 19, 4, 8, 8, 13, 8, 17, 5, 10, 7, 14, 4, 21, 3, 7, 11, 11, 11, 18, 4

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A024140, A001222, A046051, A057951, A057952, A057953, A057954, A057955, A057956, A057957, A057958, A250288, A252170, A366707, A366709, A366682.

Mathematica
```
PrimeOmega[12^Range[70]-1]
```
PARI
```
a(n)=bigomega(12^n-1)
```

a(n) = bigomega(12^n-1) = A001222(A024140(n)).

A366717 Smallest prime dividing 12^n - 1.

Original entry on oeis.org

11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5, 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Periodic with period 12, repeat of 11, 11, 11, 5, 11, 7, 11, 5, 11, 11, 11, 5.

Cf. A024140, A366718, A366719.

Table[FactorInteger[12^n - 1][[1,1]], {n,71}] (* Paul F. Marrero Romero, Oct 25 2023 *)

a(n) = A020639(A024140(n)). - Paul F. Marrero Romero, Oct 25 2023

A366718 Largest prime factor of 12^n - 1.

Original entry on oeis.org

11, 13, 157, 29, 22621, 157, 4943, 233, 80749, 22621, 266981089, 20593, 20369233, 13063, 22621, 260753, 74876782031, 80749, 29043636306420266077, 85403261, 8177824843189, 57154490053, 321218438243, 2227777, 12629757106815551, 20369233, 86769286104133

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A024140, A006530, A005420, A074477, A274906, A074479, A274907, A074249, A274908, A274909, A005422, A274910, A366707, A366708, A366709, A366710, A366711, A366717, A366720.

[Maximum(PrimeDivisors(12^n-1)): n in [1..40]];

Table[FactorInteger[12^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024140(n)).

cp's OEIS Frontend

A001021 Powers of 12.

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A016125 Expansion of 1/((1-x)*(1-12*x)).

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

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A366709 Number of divisors of 12^n-1.

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A366707 Number of distinct prime divisors of 12^n - 1.

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A366710 Sum of the divisors of 12^n-1.

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Maple

Mathematica

Formula

A366711 a(n) = phi(12^n-1), where phi is Euler's totient function (A000010).

A016125 Expansion of 1/((1-x)(1-12x)).