A038371 -id:A038371 - 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)

A119704 a(n) = number of distinct prime factors of 10^n+1 = omega(10^n+1).

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, Jun 09 2006

nonn

			a(1) = number of distinct prime factors of 11 = 1.
a(3) = number of distinct prime factors of 1001 = 3.
a(11) = 4 because 10^11+1 = 11*11*23*4093*8779 has 4 distinct factors.

Max Alekseyev, Table of n, a(n) for n = 0..331 [First 309 terms from Ray Chandler, from the Kamada link]
Makoto Kamada, Factorizations of 100...001.

Cf. A001221, A003021, A038371, A057934, A062397.

Table[Length[FactorInteger[10^n + 1]], {n, 0, 50}] (* Stefan Steinerberger, Jun 13 2006 *)
PrimeNu[10^Range[0,100]+1] (* The program will take some time to run *) (* Harvey P. Dale, Aug 27 2019 *)

a(n) = A001221(A062397(n)). - Ray Chandler, May 02 2017

More terms from Don Reble, Jun 13 2006

A344897 a(n) is the number of divisors of 10^n + 1.

Original entry on oeis.org

2, 2, 2, 8, 4, 4, 4, 4, 4, 32, 8, 24, 8, 8, 16, 128, 32, 16, 8, 4, 16, 192, 16, 32, 8, 32, 8, 128, 16, 8, 128, 4, 16, 384, 16, 32, 64, 16, 8, 768, 16, 8, 128, 16, 16, 4096, 16, 16, 512, 16, 128, 256, 16, 4, 64, 768, 32, 64, 32, 16, 64, 8, 8, 3072, 8, 64, 256, 4, 16, 1024, 2048, 8, 32, 16, 128, 2048, 64, 3072, 128, 16

Offset: 0

Seiichi Manyama, Jun 01 2021

nonn

a(n) is even because 10^n + 1 is not a square number.

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000005, A000533, A003021, A038371, A046798, A057934, A070528, A119704, A344859, A087021, A087022,

a[0] = 2; a[n_] := DivisorSigma[0, 10^n + 1]; Array[a, 60, 0] (* Amiram Eldar, Jun 01 2021 *)

PARI
```
a(n) = numdiv(10^n+1);
```

a(n) = A000005(A000533(n)).

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

A366609 Smallest prime dividing 4^n + 1.

Original entry on oeis.org

2, 5, 17, 5, 257, 5, 17, 5, 65537, 5, 17, 5, 97, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 193, 5, 17, 5, 257, 5, 17, 5, 274177, 5, 17, 5, 97, 5, 17, 5, 65537, 5, 17, 5, 257, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 449, 5, 17, 5, 97, 5, 17, 5, 59649589127497217

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..1983

Bisection of A002586.

Cf. A002586, A038371, A052539, A074476, A274903, A366605, A366606, A366607, A366608, A366670, A366671, A366719.

A185121 Smallest prime factor of 10^(2^n) + 1.

Original entry on oeis.org

11, 101, 73, 17, 353, 19841, 1265011073, 257, 10753, 1514497, 1856104284667693057, 106907803649, 458924033, 3635898263938497962802538435084289

Offset: 0

Sergio Pimentel, Jan 22 2012

10^k+1 can only be prime if k is a power of 2. So far the only known primes of this form are a(0) = 11 and a(1) = 101. [Edited by M. F. Hasler, Aug 03 2019]
a(n) >= 2^(n+1)+1; we have a(n) = 2^(n+1)+1 for n=3, n=7, and n=15.
a(14) > 10^16. - Max Alekseyev, Jun 28 2013
From the Keller link a(15)-a(20) = 65537, 8257537, 175636481, 639631361, 70254593, 167772161. - Ray Chandler, Dec 27 2013

			For n=2, a(2)=73 since 10^(2^2) + 1 = 10001 = 73 * 137.

FactorDB, Factorizations of 10^(2^n)+1
Makoto Kamada, Factorizations of 100...001
Wilfrid Keller, Prime factors of generalized Fermat numbers Fm(10) and complete factoring status

Cf. A038371, A000533, A000215, A093179, A102050

Essentially the same as A102050. - Sean A. Irvine, Feb 17 2013

Table[With[{k = 2^n}, FactorInteger[10^k + 1]][[1, 1]], {n, 0, 13, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

a(n) = factor(10^(2^n)+1)[1, 1] \\ Michel Marcus, May 30 2013

a(n) = A038371(2^n). - M. F. Hasler, Jul 30 2019

A366670 Smallest prime dividing 6^n + 1.

Original entry on oeis.org

2, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 1297, 7, 37, 7, 353, 7, 13, 7, 41, 7, 37, 7, 17, 7, 37, 7, 281, 7, 13, 7, 2753, 7, 37, 7, 577, 7, 37, 7, 17, 7, 13, 7, 89, 7, 37, 7, 193, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 41, 7, 37, 7, 4926056449, 7, 13, 7, 137

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..4095

Cf. A020639, A062394, A274904, A366630.

Cf. A002586, A366609, A366671, A038371, A366719.

Table[FactorInteger[6^n + 1][[1,1]], {n, 0, 68}] (* Paul F. Marrero Romero, Oct 17 2023 *)

a(n) = A020639(A062394(n)). - Paul F. Marrero Romero, Oct 17 2023

A366671 Smallest prime dividing 8^n + 1.

Original entry on oeis.org

2, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 769, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

a(n) = 3 if n is odd. a(n) = 5 if n == 2 (mod 4). - Robert Israel, Nov 20 2023

Max Alekseyev, Table of n, a(n) for n = 0..16383

Cf. A002586, A366609, A052536, A366655, A274905, A366670, A038371, A366719.

P1000:= mul(ithprime(i),i= 4..1000):
f:= proc(n) local t;
  if n::odd then return 3 elif n mod 4 = 2 then return 5 fi;
  t:= igcd(8^n+1,P1000);
  if t <> 1 then min(numtheory:-factorset(t)) else min(numtheory:-factorset(8^n+1)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Nov 20 2023

Table[FactorInteger[8^n + 1][[1,1]], {n, 0, 78}] (* Paul F. Marrero Romero, Oct 20 2023 *)

from sympy import primefactors
def A366671(n): return min(primefactors((1<<3*n)+1)) # Chai Wah Wu, Oct 16 2023

a(n) = A020639(A062395(n)). - Paul F. Marrero Romero, Oct 20 2023

a(n) = A002586(3*n) for n >= 1. - Robert Israel, Nov 20 2023

A262083 Smallest possible prime factor of 10^k+n for any k.

Original entry on oeis.org

2, 7, 2, 7, 2, 3, 2, 17, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 11, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 13, 2, 7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 7, 2, 17, 2, 3, 2, 7, 2, 7, 2, 3, 2, 7, 2, 5, 2, 3, 2, 7, 2, 7, 2, 3, 2, 5, 2, 7, 2, 3, 2, 17, 2, 7, 2, 3, 2, 7, 2, 7, 2

Offset: 0

Sergio Pimentel, Sep 10 2015

nonn

Is this sequence bounded?  What are the records for a(n)?
From Robert G. Wilson v, Sep 13 2015: (Start)
First occurrence of the i-th prime: 0, 5, 15, 1, 21, 49, 7, 357, 24871, 364021, ..., .
a(n) = 2 when n == 0 (mod 2),
a(n) = 3 when n == 5 (mod 6),
a(n) = 5 when n == 15 or 25 (mod 30),
a(n) = 7 when n == 1, 3, 9, 13, 19, 27, 31, 33, 37, 39, 43, 51, 57, 61, 67, 69, 73, 79, 81, 87, 93, 97, 99, 103, 109, 111, 117, 121, 123, 127, 129, 139, 141, 151, 153, 157, 159, 163, 169, 171, 177, 181, 183, 187, 193, 199, 201 or 207 (mod 210),
a(n) = 11 when n = 21, 133, 441, 483, 637, 903, 1057, 1099, 1407, 1519, 1561, 1827, 1869, 1981, 2023 or 2289 (mod 2310),
a(n) = 13 when n = 49, 147, 217, 231, 259, 399, 469, 511, 651, 679, 693, 763, 777, 861, 987, 1141, 1197, (413 terms missing), 29883 or 29953, ... (mod 30030),
a(n) = 17 when n = 7, 63, 91, 189, 273, 301, 343, 427, 553, 567, 609, 721, 819, 847, 889, 931, 973, 1029, (8044 terms missing), 510349 or 510447 (mod 510510),
a(n) = 19 when n = 357, 1071, 2737, 3451, 6069, 6307, 8211, 9163, 9639, 10353, 12019, 12733, 13447, 13923, 15351, 15589, 17017, 17493, 18207, ... (mod 9699690),
a(n) = 23 when n = 24871, 47481, 74613, 88179, 92701, 106267, 133399, 142443, 160531, 187663, 201229, 210273, 223839, 250971, 264537, 309757, ... (mod 223092870),
a(n) = 29 when n = 364021, 988057, ... (mod 6469693230), etc.
To the question if this sequence is 'bounded', I would answer no.
(End)
For complete lists of when a(n) < 19, see Wilson's Congruencies a-file. - Danny Rorabaugh, Oct 08 2015

			a(1) = 7 since 10^k+1 is not divisible by 2,3 or 5 for all k but is divisible by 7 when k = 3 (i.e., 1001 = 7*11*13).

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Robert G. Wilson v, Congruencies for A262083

Cf. A000533, A003617, A038371, A185121.

p = Prime@ Range@ 25; f[n_] := Block[{k = 1, lst = {}}, While[k < 25, AppendTo[lst, Position[ Mod[ PowerMod[10, k, p] + n, p] 0, 1, 1][[1, 1]]]; k++]; lst = Union@ lst; Prime@ lst[[1]]]; Array[f, 101, 0] (* Robert G. Wilson v, Sep 13 2015 *)

More terms from Robert G. Wilson v, Sep 13 2015

A309358 Numbers k such that 10^k + 1 is a semiprime.

Original entry on oeis.org

4, 5, 6, 7, 8, 19, 31, 53, 67, 293, 586, 641, 922, 2137, 3011

Offset: 1

Hugo Pfoertner, Jul 29 2019

a(16) > 12000.
10^k + 1 is composite unless k is a power of 2, and it can be conjectured that it is composite for all k > 2, cf. A038371 and A185121. - M. F. Hasler, Jul 30 2019
Suppose k is odd. Then k is a term if and only if (10^k+1)/11 is prime. - Chai Wah Wu, Jul 31 2019

			a(1) = 4 because 10^4 + 1 = 10001 = 73*137.

Cf. A003021, A038371, A046413, A057934, A062397, A092559.

Odd terms in sequence: A001562.

IsSemiprime:=func; [n: n in [2..200] | IsSemiprime(s) where s is 10^n+1]; // Vincenzo Librandi, Jul 31 2019

Mathematica

Select[Range[200], Plus@@Last/@FactorInteger[10^# + 1] == 2 &] (* Vincenzo Librandi, Jul 31 2019 *)

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

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A119704 a(n) = number of distinct prime factors of 10^n+1 = omega(10^n+1).

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A344897 a(n) is the number of divisors of 10^n + 1.

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A366719 Smallest prime dividing 12^n + 1.

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A366609 Smallest prime dividing 4^n + 1.

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A185121 Smallest prime factor of 10^(2^n) + 1.

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A366670 Smallest prime dividing 6^n + 1.

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A366671 Smallest prime dividing 8^n + 1.

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