A178248 -id:A178248 - cp's OEIS frontend

A062397 a(n) = 10^n + 1.

2, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001, 1000000000001, 10000000000001, 100000000000001, 1000000000000001, 10000000000000001, 100000000000000001

Offset: 0

Henry Bottomley, Jun 22 2001

The first three terms (indices 0, 1 and 2) are the only known primes. Moreover, the terms not of the form a(2^k) are all composite, except for a(0). Indeed, for all n >= 0, a(2n+1) is divisible by 11, a(4n+2) is divisible by 101, a(8n+4) is divisible by 73, a(16n+8) is divisible by 17, a(32n+16) is divisible by 353, a(64n+32) is divisible by 19841, etc. - M. F. Hasler, Nov 03 2018 [Edited based on the comment by Jeppe Stig Nielsen, Oct 17 2019]
This sequence also results when each term is generated by converting the previous term into a Roman numeral, then replacing each letter with its corresponding decimal value, provided that the vinculum is used and numerals are written in a specific way for integers greater than 3999, e.g., IV with a vinculum over the I and V for 4000. - Jamie Robert Creasey, Apr 14 2021
By Mihăilescu's theorem, a(n) can never be a perfect power (see "Catalan's conjecture" in Links). - Marco Ripà, Mar 10 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wikipedia, Catalan's conjecture.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Except for the initial term, essentially the same as A000533. Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081 for numbers one more than powers, i.e., this sequence translated from base n (> 2) to base 10.
Cf. A002283, A011557.
Cf. A038371 (smallest prime factor), A185121.

[10^n + 1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

LinearRecurrence[{11, -10},{2, 11},18] (* Ray Chandler, Aug 26 2015 *)
10^Range[0,20]+1 (* Harvey P. Dale, Jan 21 2020 *)

a(n)=10^n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 10*a(n-1) - 9 = A011557(n) + 1 = A002283(n) + 2.
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-10*x).
E.g.f.: exp(x) + exp(10*x). (End)

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

Original entry on oeis.org

2, 12, 122, 1332, 14642, 161052, 1771562, 19487172, 214358882, 2357947692, 25937424602, 285311670612, 3138428376722, 34522712143932, 379749833583242, 4177248169415652, 45949729863572162, 505447028499293772

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Sequences of the form m^n + 1: A000012 (m=0), A007395 (m=1), A000051 (m=2), A034472 (m=3), A052539 (m=4), A034474 (m=5), A062394 (m=6), A034491 (m=7), A062395 (m=8), A062396 (m=9), A062397 (m=10), this sequence (m=11), A178248 (m=12), A141012 (m=13), A228081 (m=64).

Cf. A001020.

[11^n +1: n in [0..30]]; // G. C. Greubel, Mar 11 2023

LinearRecurrence[{12,-11},{2,12},18] (* Ray Chandler, Aug 26 2015 *)
11^Range[0,30]+1 (* G. C. Greubel, Mar 11 2023 *)

a(n)=11^n+1 \\ Charles R Greathouse IV, Sep 24 2015

[sigma(11,n)for n in range(0,18)] # - Zerinvary Lajos, Jun 04 2009

From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-11*x).
E.g.f.: exp(x) + exp(11*x). (End)
From G. C. Greubel, Mar 11 2023: (Start)
a(n) = 11*a(n-1) - 10.
a(n) = A001020(n) + 1. (End)

A366712 Number of distinct prime divisors of 12^n + 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 3, 2, 4, 4, 4, 7, 2, 3, 5, 4, 6, 8, 6, 4, 6, 6, 5, 10, 3, 3, 5, 3, 2, 9, 4, 5, 7, 4, 5, 10, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 8, 9, 5, 5, 10, 7, 9, 10, 7, 6, 10, 5, 6, 14, 5, 8, 11, 4, 5, 12, 9, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15, 3

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A001221, A046799, A119704, A366713, A366714, A366707, A366686.

for(n = 0, 100, print1(omega(12^n + 1), ", "))

a(n) = omega(12^n+1) = A001221(A178248(n)).

A366714 Number of divisors of 12^n+1.

Original entry on oeis.org

2, 2, 4, 8, 4, 4, 8, 8, 8, 32, 12, 4, 16, 24, 16, 128, 4, 8, 32, 16, 64, 384, 64, 16, 64, 64, 32, 1024, 8, 8, 48, 8, 4, 512, 16, 32, 128, 16, 32, 1536, 16, 32, 64, 32, 16, 4096, 8, 32, 32, 32, 512, 512, 32, 32, 1024, 128, 512, 1536, 192, 64, 1024, 32, 64

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

			a(4)=4 because 12^4+1 has divisors {1, 89, 233, 20737}.

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A000005, A046798, A344897, A366709, A366712, A366713, A366715, A366716, A366719, A366720, A366688.

a:=n->numtheory[tau](12^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 12^Range[0, 70] + 1] (* Paolo Xausa, Apr 20 2025 *)

PARI
```
a(n) = numdiv(12^n+1);
```

a(n) = sigma0(12^n+1) = A000005(A178248(n)).

A366716 a(n) = phi(12^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 12, 112, 1296, 20416, 229680, 2306304, 32916240, 400515072, 3863116800, 47825825600, 685853880624, 8732596764672, 97509650382144, 990242755633152, 11148606564480000, 184883057981234176, 2047145911595946000, 20281543142263603200, 294779525244632305920

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A000010, A366711, A366712, A366713, A366714, A366715, A366719, A366720, A366690.

EulerPhi[12^Range[0,19] + 1] (* Paul F. Marrero Romero, Oct 27 2023 *)

PARI
```
{a(n) = eulerphi(12^n+1)}
```

a(n) = A000010(A178248(n)). - Paul F. Marrero Romero, Oct 27 2023

A366715 Sum of the divisors of 12^n+1.

Original entry on oeis.org

3, 14, 180, 2240, 21060, 267988, 3706920, 38773952, 459970056, 6692483840, 79425033660, 800162860756, 9101898907920, 117326869641600, 1596198064568400, 20655000929239040, 184885459808838660, 2390210102271311936, 33504016991491136160, 344201347103878781440

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

			a(4)=21060 because 12^4+1 has divisors {1, 89, 233, 20737}.

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A000203, A366710, A366712, A366713, A366714, A366716, A366719, A366720, A366689.

a:=n->numtheory[sigma](12^n+1):
seq(a(n), n=0..100);

a(n) = sigma(12^n+1) = A000203(A178248(n)).

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

Original entry on oeis.org

0, 11, 143, 1727, 20735, 248831, 2985983, 35831807, 429981695, 5159780351, 61917364223, 743008370687, 8916100448255, 106993205379071, 1283918464548863, 15407021574586367, 184884258895036415

Offset: 0

N. J. A. Sloane

In base 12 these are 0, B, BB, BBB, ... . - David Rabahy, Dec 12 2016

Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A001021, A016125, A178248.

Cf. Similar sequences of the type k^n-1: A000004 (k=1), A000225 (k=2), A024023 (k=3), A024036 (k=4), A024049 (k=5), A024062 (k=6), A024075 (k=7), A024088 (k=8), A024101 (k=9), A002283 (k=10), A024127 (k=11), this sequence (k=12).

12^Range[0,20]-1 (* or *) LinearRecurrence[{13,-12},{0,11},20] (* Harvey P. Dale, Feb 01 2019 *)

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-12*x) - 1/(1-x).
E.g.f.: exp(12*x) - exp(x). (End)
a(n) = 12*a(n-1) + 11 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{i=1..n} 11^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
From Elmo R. Oliveira, Dec 15 2023: (Start)
a(n) = 13*a(n-1) - 12*a(n-2) for n>1.
a(n) = A001021(n)-1 = A178248(n)-2.
a(n) = 11*(A016125(n) - 1)/12. (End)

A366719 Smallest prime dividing 12^n + 1.

Original entry on oeis.org

2, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 89, 13, 5, 7, 153953, 13, 5, 13, 41, 7, 5, 13, 17, 13, 5, 7, 89, 13, 5, 13, 769, 7, 5, 13, 89, 13, 5, 7, 17, 13, 5, 13, 89, 7, 5, 13, 7489, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 41, 13, 5, 7, 36097, 13, 5, 13, 89, 7

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..511

Cf. A002586, A038371, A178248, A366712, A366713, A366714, A366715, A366716, A366717, A366720.

Table[FactorInteger[12^n + 1][[1,1]], {n, 0, 69}] (* Paul F. Marrero Romero, Oct 25 2023 *)

a(n) = A020639(A178248(n)). - Paul F. Marrero Romero, Oct 25 2023

A366713 Number of prime factors of 12^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 4, 2, 4, 5, 4, 7, 2, 3, 5, 4, 6, 9, 6, 4, 6, 6, 5, 10, 3, 3, 6, 3, 2, 9, 4, 5, 7, 4, 5, 11, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 10, 9, 5, 5, 10, 7, 9, 11, 8, 6, 10, 5, 6, 15, 5, 9, 11, 4, 5, 12, 10, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A001222, A054992, A057934, A057935, A057936, A057937, A057938, A057939, A057940, A057941, A366712, A366714, A366715, A366716, A366719, A366720, A366708, A366687.

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

a(n) = bigomega(12^n+1) = A001222(A178248(n)).

A366720 Largest prime factor of 12^n+1.

Original entry on oeis.org

2, 13, 29, 19, 233, 19141, 20593, 13063, 260753, 1801, 85403261, 57154490053, 2227777, 222379, 13156924369, 35671, 1200913648289, 66900193189411, 122138321401, 905265296671, 67657441, 1885339, 68368660537, 49489630860836437, 592734049, 438472201

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A006530, A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590, A366712, A366713, A366714, A366715, A366716, A366717, A366718, A366719, A324941.

Table[FactorInteger[12^n + 1][[-1, 1]], {n, 0, 20}]

a(n) = A006530(A178248(n)). - Paul F. Marrero Romero, Dec 07 2023

cp's OEIS Frontend

A062397 a(n) = 10^n + 1.

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Magma

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

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A366712 Number of distinct prime divisors of 12^n + 1.

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A366714 Number of divisors of 12^n+1.

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A366716 a(n) = phi(12^n+1), where phi is Euler's totient function (A000010).

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A366715 Sum of the divisors of 12^n+1.

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

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A366719 Smallest prime dividing 12^n + 1.

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