A074476 -id:A074476 - cp's OEIS frontend

A227575 Largest prime factor of 7^n + 1.

2, 2, 5, 43, 1201, 191, 181, 911, 169553, 117307, 4021, 10746341, 1201, 228511817, 13564461457, 6568801, 47072139617, 29078814248401, 13841169553, 4058036683, 810221830361, 309079, 83960385389, 3421093417510114543, 33232924804801, 79787519018560501

Offset: 0

Michel Marcus, Aug 22 2013

nonn

			7^12 + 1 = 2*73*193*409*1201, so a(12) = 1201.

Table of n, a(n) for n = 0..387
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006530, A034491.

Cf. A002587, A074249, A074476, A074478.

[Maximum(PrimeDivisors(7^n+1)): n in [0..30]]; // Bruno Berselli, Aug 23 2013

Table[FactorInteger[7^n + 1][[-1, 1]], {n, 0, 30}] (* Bruno Berselli, Aug 23 2013 *)

a(n) = f = factor(7^n + 1); f[#f~, 1]; \\ Michel Marcus, Aug 22 2013

a(n) = A006530(A034491(n)). - Vincenzo Librandi, Jul 12 2016

Terms to a(100) in b-file from Vincenzo Librandi, Jul 12 2016
a(101)-a(372) in b-file from Amiram Eldar, Feb 02 2020
a(373)-a(387) in b-file from Max Alekseyev, Apr 25 2022, Aug 30 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.

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

A063271 Largest prime factor of 9^(2n)+1 (A063270).

Original entry on oeis.org

2, 41, 193, 6481, 21523361, 42521761, 769, 647753, 926510094425921, 282429005041, 128653413121, 56625998353, 24127552321, 37644053098601, 36214795668330833, 42521761, 1716841910146256242328924544641, 3833564416504313, 56227703611393, 278733912072436804273

Offset: 0

Jason Earls, Jul 12 2001

nonn

Daniel Suteu and Harry J. Smith, Table of n, a(n) for n = 0..172 (terms a(0)..a(50) from Harry J. Smith)

Cf. A006530, A063270, A074476, A002592.

Table[FactorInteger[9^(2n)+1][[-1,1]],{n,0,20}] (* Harvey P. Dale, Jan 07 2013 *)

a(n)={vecmax(factor(9^(2*n) + 1)[,1])} \\ Harry J. Smith, Aug 20 2009

a(n) = A006530(A063270(n)) = A002592(2*n) = A074476(4*n). - Daniel Suteu, May 26 2022

Definition corrected by Harry J. Smith, Aug 20 2009

A189240 Least number k such that 2kn + 1 is a prime dividing 3^n + 1.

Original entry on oeis.org

1, 1, 5, 6, 6, 39, 1, 1, 59, 3, 270, 15330, 1, 1, 672605, 3, 2, 75, 1, 1, 125, 511647711, 2, 3, 1, 360, 7691, 9, 796056, 111, 14476720225405, 1, 14064, 5355114024, 90, 249, 69757, 1, 180

Offset: 2

Michel Lagneau, Apr 19 2011

nonn

The smallest prime factor of 3^n+1 of the form 2k*n+1 is A189241(n).

			a(4) = 5 because 3^4+1 = 2*41 => the smallest prime divisor of the form  2k*n+1 is 41 = 2*5*4+1.

Amiram Eldar, Table of n, a(n) for n = 2..658

Cf. A189241, A074476 (largest prime factor of 3^n + 1)

Table[p=First/@FactorInteger[3^n+1]; (Select[p, Mod[#1, n] == 1 &, 1][[1]]
  - 1)/(2n), {n, 2, 40}]

a(n)=forstep(K=2*n+1,3^n+1,2*n,if(Mod(3,K)^n==0,return((k-1)/2/n))) \\ Charles R Greathouse IV, May 15 2013

A189241 Smallest prime factor of 3^n+1 having the form 2kn+1.

Original entry on oeis.org

5, 7, 41, 61, 73, 547, 17, 19, 1181, 67, 6481, 398581, 29, 31, 21523361, 103, 73, 2851, 41, 43, 5501, 23535794707, 97, 151, 53, 19441, 430697, 523, 47763361, 6883, 926510094425921, 67, 956353, 374857981681, 6481, 18427, 5301533, 79, 14401

Offset: 2

Michel Lagneau, Apr 19 2011

nonn

The values of k are in A189240.

			a(4) = 41 because 3^4 + 1 = 2 * 41 ; the smallest prime divisor of the form  2*k*n+1 is 41 = 2*5*4+1.

Amiram Eldar, Table of n, a(n) for n = 2..658

Cf. A189240, A074476 (largest prime factor of 3^n + 1).

Table[p=First/@FactorInteger[3^n+1]; Select[p, Mod[#1, n] == 1 &, 1][[1]],
  {n, 2, 40}]

A324941 Largest prime factor of 17^n + 1.

Original entry on oeis.org

2, 3, 29, 13, 41761, 101, 83233, 22796593, 184417, 5653, 63541, 87415373, 72337, 2001793, 100688449, 238212511, 52548582913, 45957792327018709121, 382069, 20352763, 1186844128302568601, 88109799136087, 6901823633, 1109309383381084655697725873, 48661191868691111041

Offset: 0

Vincenzo Librandi, Apr 05 2019

nonn

Max Alekseyev, Table of n, a(n) for n = 0..211

Cf. A006530, A224384.

Cf. A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590.

[Maximum(PrimeDivisors(17^n + 1)): n in [0..40]];

Table[FactorInteger[17^n + 1] [[-1,1]], {n, 0, 30}]

a(n) = vecmax(factor(17^n+1)[, 1]); \\ Jinyuan Wang, Apr 05 2019

a(n) = A006530(A224384(n)).

A167205 a(n) = (3^n+1)/(3-(-1)^n).

Original entry on oeis.org

1, 1, 5, 7, 41, 61, 365, 547, 3281, 4921, 29525, 44287, 265721, 398581, 2391485, 3587227, 21523361, 32285041, 193710245, 290565367, 1743392201, 2615088301, 15690529805, 23535794707, 141214768241, 211822152361, 1270932914165

Offset: 0

Jaume Oliver Lafont, Oct 30 2009

This sequence is (3^n + 1) divided by the highest possible power of 2, which is 4 for odd n and 2 for even n. It is never divisible by 8 or any higher power of 2, which implies Levi ben Gerson's observation that (3^n + 1 = 2^k) has no solution for n > 1. Cf. the comments and links to A235365. - Joe Slater, Apr 02 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-9)

Cf. A007051, A074476, A235365.

List([0..27],n->(3^n+1)/(3-(-1)^n)); # Muniru A Asiru, Mar 05 2018

a:=n->(3^n+1)/(3-(-1)^n): seq(a(n),n=0..27); # Muniru A Asiru, Mar 05 2018

CoefficientList[Series[(1+x-5x^2-3x^3)/((1+x)(1-x)(1+3x)(1-3x)), {x,0,30}],x] (* or *) LinearRecurrence[{0,10,0,-9},{1,1,5,7},30] (* Harvey P. Dale, Apr 25 2011 *)

a(n) = (3^n+1)/(3-(-1)^n); \\ Altug Alkan, Mar 05 2018

a(n) = 10*a(n-2) - 9*a(n-4).
G.f.: (1 + x - 5*x^2 - 3*x^3)/((1+x)*(1-x)*(1+3*x)*(1-3*x)).
a(n) = numerator((1/4)^n + (3/4)^n), n > 0.

cp's OEIS Frontend

A227575 Largest prime factor of 7^n + 1.

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A366609 Smallest prime dividing 4^n + 1.

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A366720 Largest prime factor of 12^n+1.

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A063271 Largest prime factor of 9^(2n)+1 (A063270).

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A189240 Least number k such that 2*k*n + 1 is a prime dividing 3^n + 1.

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A189241 Smallest prime factor of 3^n+1 having the form 2*k*n+1.

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Examples

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Mathematica

A324941 Largest prime factor of 17^n + 1.

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Magma

Mathematica

PARI

Formula

A167205 a(n) = (3^n+1)/(3-(-1)^n).

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A189240 Least number k such that 2kn + 1 is a prime dividing 3^n + 1.

A189241 Smallest prime factor of 3^n+1 having the form 2kn+1.