A002587 -id:A002587 - cp's OEIS frontend

A274905 Largest prime factor of 8^n + 1.

2, 3, 13, 19, 241, 331, 109, 5419, 673, 87211, 1321, 20857, 38737, 22366891, 14449, 18837001, 22253377, 43691, 279073, 160465489, 4562284561, 77158673929, 4327489, 168749965921, 487824887233, 1133836730401, 21841, 272010961, 88959882481, 96076791871613611

Offset: 0

Vincenzo Librandi, Jul 11 2016

nonn

Table of n, a(n) for n = 0..502
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A002587, A006530, A062395.

Cf. similar sequences listed in A274903.

[Maximum(PrimeDivisors(8^n+1)): n in [0..40]];

Maple
```
8^4 + 1 = 4097 = 17*241, so a(4) = 241.
```

Table[FactorInteger[8^n + 1][[-1, 1]], {n, 0, 40}]

a(n) = A006530(A062395(n)). - Michel Marcus, Jul 11 2016

a(n) = A002587(3*n). - Amiram Eldar, Feb 02 2020

Terms to a(100) in b-file from Vincenzo Librandi, Jul 12 2016
a(101)-a(354) in b-file from Amiram Eldar, Feb 02 2020
a(355)-a(502) in b-file from Max Alekseyev, May 28 2022, Sep 06 2022, Feb 25 2023

A227575 Largest prime factor of 7^n + 1.

Original entry on oeis.org

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

A002590 Largest prime factor of 16^n + 1.

Original entry on oeis.org

2, 17, 257, 241, 65537, 61681, 673, 15790321, 6700417, 38737, 4278255361, 2931542417, 22253377, 308761441, 54410972897, 4562284561, 67280421310721, 2879347902817, 487824887233, 24517014940753, 44479210368001, 88959882481

Offset: 0

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 88.
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).

Max Alekseyev, Table of n, a(n) for n = 0..291 (terms 0..256 from Sean A. Irvine, term 277 from Tyler Busby)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Join[{1},Table[FactorInteger[16^n+1][[-1,1]],{n,25}]] (* Harvey P. Dale, Feb 26 2012 *)

a(n) = A002587(4*n).

More terms from Don Reble, Nov 14 2006

a(0) corrected by Sean A. Irvine, Apr 20 2014

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

A283657 Numbers m such that 2^m + 1 has at most 2 distinct prime factors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 19, 20, 23, 28, 31, 32, 40, 43, 61, 64, 79, 92, 101, 104, 127, 128, 148, 167, 191, 199, 256, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479

Offset: 1

Vladimir Shevelev, Mar 13 2017

nonn

Using comment in A283364, note that if a(n) is odd > 9, then it is prime.
503 <= a(41) <= 596. - Robert Israel, Mar 13 2017
Could (4^p + 1)/5^t be prime, where p is prime, 5^t is the highest power of 5 dividing 4^p + 1, other than for p=2, 3 and 5? - Vladimir Shevelev, Mar 14 2017
In his message to seqfans from Mar 15 2017, Jack Brennen beautifully proved that there are no more primes of such form. From his proof one can see also that there are no terms of the form 2*p > 10 in the sequence. - Vladimir Shevelev, Mar 15 2017
Where A046799(n)=2. - Robert G. Wilson v, Mar 15 2017
From Giuseppe Coppoletta, May 16 2017: (Start)
The only terms that are not in A066263 are those m giving 2^m + 1 = prime (i.e. m = 0 and any number m such that 2^m + 1 is a Fermat prime) and the values of m giving 2^m + 1 = power of a prime, giving m = 3 as the only possible case (by Mihăilescu-Catalan's result, see links).
For the relation with Fermat numbers and for other possible terms to check, see comments in A073936 and A066263.
All terms after a(59) refer to probabilistic primality tests for 2^a(n) + 1  (see Caldwell's link for the list of the largest certified Wagstaff primes).
After a(65), the values 267017, 269987, 374321, 986191, 4031399 and 4101572 are also terms, but there still remains the remote possibility of some gaps in between. In addition, 13347311 and 13372531 are also terms, but possibly much further along in the numbering (see comments in A000978).
(End).

			0 is a term as 2^0 + 1 = 2 is a prime.
10 is a term as 2^10 + 1 = 5^2 * 41.
14 is not a term as 2^14 + 1 = 5 * 29 * 113.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..65
Jack Brennen, Primes of the form (4^p+1)/5^t, Seqfan (Mar 15 2017).
C. Caldwell's The Top Twenty Wagstaff primes.
Mersennewiki, Factorizations Of Cunningham Numbers C+(2,n) (tables).
Samuel S. Wagstaff, The Cunningham Project.
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Eric Weisstein's World of Mathematics, Zsigmondy Theorem.

Cf. A002587, A046799, A283364, A073936, A000978, A127317, A092559, A019434, A066263.

Contains 4*A057182.

# this uses A002587[i] for i<=500, e.g., from the b-file for that sequence
count:= 0:
for i from 0 to 500 do
  m:= 0;
  r:= (2^i+1);
  if i::odd then
    m:= 1;
    r:= r/3^padic:-ordp(r,3);
  elif i > 2 then
    q:= max(numtheory:-factorset(i));
    if q > 2 then
      m:= 1;
      r:= r/B[i/q]^padic:-ordp(r,A002587[i/q]);
    fi
  fi;
  if r mod B[i] = 0 then m:= m+1;
      j:= padic:-ordp(r, A002587[i]);
      r:= r/B[i]^j;
  fi;
  mmax:= m;
  if isprime(r) then m:= m+1; mmax:= m
  elif r > 1 then mmax:= m+2
  fi;
  if mmax <= 2 or (m <= 1 and m + nops(numtheory:-factorset(r)) <= 2) then
       count:= count+1;
     A[count]:= i;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Mar 13 2017

Select[Range[0, 313], PrimeNu[2^# + 1]<3 &] (* Indranil Ghosh, Mar 13 2017 *)

for(n=0, 313, if(omega(2^n + 1)<3, print1(n,", "))) \\ Indranil Ghosh, Mar 13 2017

a(16)-a(38) from Peter J. C. Moses, Mar 13 2017
a(39)-a(40) from Robert Israel, Mar 13 2017
a(41)-a(65) from Giuseppe Coppoletta, May 08 2017

A271314 Largest prime factor of the n-th Jacobsthal number, A001045(n).

Original entry on oeis.org

3, 5, 11, 7, 43, 17, 19, 31, 683, 13, 2731, 127, 331, 257, 43691, 73, 174763, 41, 5419, 683, 2796203, 241, 4051, 8191, 87211, 127, 3033169, 331, 715827883, 65537, 20857, 131071, 86171, 109, 25781083, 524287, 22366891, 61681, 8831418697, 5419, 2932031007403, 2113, 18837001

Offset: 3

Altug Alkan, Apr 03 2016

nonn

a(22) = 683 is the first repeated term in this sequence. Note that a(n+2) = A129738(n), for n < 20.

			a(6) = 7 because A001045(6) = 21 = 3*7.

Amiram Eldar, Table of n, a(n) for n = 3..1122 (terms 3..988 from Charles R Greathouse IV)

Essentially a combination of A005420 and A002587.

Cf. A001045, A006530, A060385, A129738, A264137.

FactorInteger[#][[-1, 1]] & /@ Take[#, -(Length@ # - 3)] &@ CoefficientList[Series[x/(1 - x - 2 x^2), {x, 0, 45}], x] (* Michael De Vlieger, Apr 04 2016, after Robert G. Wilson v at A001045 *)

a001045(n) = (2^n - (-1)^n) / 3;
a(n) = vecmax(factor(a001045(n))[,1]);

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)).

A337431 Numbers k such that the largest prime factor of 2^k - 1 is greater than the largest prime factor of 2^k + 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 14, 17, 19, 26, 27, 31, 33, 34, 35, 37, 46, 49, 51, 59, 61, 62, 65, 69, 74, 77, 78, 82, 83, 86, 89, 93, 97, 103, 107, 115, 118, 121, 122, 123, 127, 129, 130, 131, 133, 137, 141, 142, 143, 144, 145, 147, 150, 153, 154, 165, 166, 169, 170, 174, 175

Offset: 1

Hugo Pfoertner, Sep 23 2020

nonn

Cf. A002587, A005420, A337430 (complement).

for(n=2,175,my(p=vecmax(factor(2^n-1)[,1]),q=vecmax(factor(2^n+1)[,1]));if(p>q,print1(n,", ")))

A070314 a(n) = P(n!+1)-P(2^n+1) where P(x) is the largest prime factor in x.

Original entry on oeis.org

0, -1, -2, 4, -12, 0, 90, 28, 404, 250, 329850, 39916118, 2834088, 75021616, 3790360374, 46271010, 993974, 956666, 123610842, 1713311273189068, 117876621366, 2703875810364, 93799610095767534, 148139754734068388, 765041185860961083618, 38681321803817920155550

Offset: 0

Benoit Cloitre, May 12 2002

sign

Is it always true that a(n) > 0 for n > 5? More generally, if m is an integer > 2, is there always an integer f(m) such that P(n!+1) > P(m^n+1) for n > f(m) (it seems that f(2) = 5, f(3) = 7, f(4) = 17, ...).

Amiram Eldar, Table of n, a(n) for n = 0..139

Cf. A002583, A002587, A006530.

gpf[n_] := FactorInteger[n][[-1,1]]; a[n_] := gpf[n!+1] - gpf[2^n+1]; Array[a, 26, 0] (* Amiram Eldar, Apr 23 2025 *)

a(n) = A002583(n) - A002587(n). - Amiram Eldar, Apr 23 2025

Offset changed to 0 and a(0) prepended by Amiram Eldar, Apr 23 2025

A158895 A list of primes written in order of their first appearance in a table of prime factorizations of 2^k+1, k=1,2,... .

Original entry on oeis.org

3, 5, 17, 11, 13, 43, 257, 19, 41, 683, 241, 2731, 29, 113, 331, 65537, 43691, 37, 109, 174763, 61681, 5419, 397, 2113, 2796203, 97, 673, 251, 4051, 53, 157, 1613, 87211, 15790321, 59, 3033169, 61, 1321, 715827883

Offset: 1

Martin Griffiths, Mar 29 2009

This sequence has the property that if a(n) appears first in the table as a prime factor of 2^m+1 for some m then a(n)=2*k*m+1 for some k.

When, for some m, 2^m+1 has more than one prime factor appearing in the table for the first time, we adopt the convention of entering them in ascending order. For example, the entries ..., 29, 113, ... both arise from 2^14+1.

			2^1+1=3, 2^2+1=5, 2^3+1=3^2 and 2^4+1=17. Thus a(1)=3, a(2)=5 and a(3)=17, on noting that 2^3+1 contributes no new prime factors.

Harvey P. Dale and Charles R Greathouse IV, Table of n, a(n) for n = 1..4017 (first 650 terms from Dale)

Subsequence of A001269.

Cf. A002587, A066845, A108974.

DeleteDuplicates[Flatten[Table[Transpose[FactorInteger[2^k+1]][[1]],{k,50}]]] (* Harvey P. Dale, Mar 30 2014 *)

lista(n)=prs = Set(); for (k=1, n, f = factor(2^k+1); for (i=1, length(f~), onef = f[i,1]; if (! setsearch(prs, onef), print1(onef, ", "); prs = setunion(prs, Set(onef));););); \\ Michel Marcus, Apr 18 2013

G=1; for(n=1,500, g=gcd(f=2^n+1,G); while(g>1, g=gcd(g,f/=g)); f=factor(f)[,1]; if(#f, for(i=1,#f, print1(f[i]", ")); G*=factorback(f))) \\ Charles R Greathouse IV, Jan 03 2018

cp's OEIS Frontend

A274905 Largest prime factor of 8^n + 1.

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A227575 Largest prime factor of 7^n + 1.

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A002590 Largest prime factor of 16^n + 1.

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A366720 Largest prime factor of 12^n+1.

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A283657 Numbers m such that 2^m + 1 has at most 2 distinct prime factors.

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Maple

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Extensions

A271314 Largest prime factor of the n-th Jacobsthal number, A001045(n).

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A324941 Largest prime factor of 17^n + 1.

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Magma

Mathematica

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