A274906 -id:A274906 - cp's OEIS frontend

A274910 Largest prime factor of 11^n - 1.

5, 5, 19, 61, 3221, 37, 45319, 7321, 1772893, 13421, 1806113, 1117, 3158528101, 1623931, 195019441, 6304673, 50544702849929377, 1772893, 6115909044841454629, 212601841, 45319, 1806113, 3740221981231, 20113, 1856458657451, 3158528101, 5559917315850179173

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			11^4 - 1 = 14640 = 2^4*3*5*61, so a(4) = 61.

Table of n, a(n) for n = 1..316
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A006530, A024127.

Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(11^n-1)): n in [1..40]];

Table[FactorInteger[11^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024127(n)).

Terms to a(70) in b-file from Vincenzo Librandi, Jul 13 2016
a(71)-a(306) in b-file from Amiram Eldar, Feb 08 2020
a(307)-a(316) in b-file from Max Alekseyev, Apr 25 2022, Oct 26 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)).

A358699 a(n) is the largest prime factor of 2^(prime(n) - 1) - 1.

Original entry on oeis.org

3, 5, 7, 31, 13, 257, 73, 683, 127, 331, 109, 61681, 5419, 2796203, 8191, 3033169, 1321, 599479, 122921, 38737, 22366891, 8831418697, 2931542417, 22253377, 268501, 131071, 28059810762433, 279073, 54410972897, 77158673929, 145295143558111, 2879347902817, 10052678938039

Offset: 2

Hugo Pfoertner, Nov 27 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 2..197 (terms 2..120 from Hugo Pfoertner)

Cf. A005420, A006093, A006530, A061286, A071243, A086019, A098102.

Subsequence of A005420 and of A274906.

forprime (p=3, 140, my(f=factor(2^(p-1)-1)); print1(f[#f[,1],1],", "))

from sympy import primefactors, sieve
def A358699(n): return primefactors(2**(sieve[n]-1)-1)[-1] # Karl-Heinz Hofmann, Nov 28 2022

a(n) = A006530(A098102(n)). - Michel Marcus, Nov 28 2022

a(n) = A005420(A006093(n)). - Amiram Eldar, Dec 01 2022

A187063 Numbers of the form (4^k - 1)/3 whose greatest prime divisor is of the form 2^q - 1 or 2^q + 1.

Original entry on oeis.org

5, 21, 85, 341, 5461, 21845, 22369621, 89478485, 1431655765, 5726623061, 91625968981, 1501199875790165, 1537228672809129301, 98382635059784275285, 1690200800304305868662270940501, 1772303994379887830538409413707126101

Offset: 1

Michel Lagneau, Mar 03 2011

nonn

The binary expansion of (4^k-1)/3 has no consecutive equal binary digits.

The corresponding values of k are 2, 3, 4, 5, 7, 8, 13, 14, 16, 17, 19, 26, 31, 34, 51, 61, 62, 89, 107, 122, 127, 178, 214, 254, 521, ... - Amiram Eldar, Mar 02 2020

			(4^6-1)/3 = 1365 = 3 * 5 * 7 * 13 is not in the sequence because  13 is not of the form 2^q +/- 1 ;
(4^16-1)/3 = 1431655765 = 5 * 17 * 257 * 65537 and 65537 = 2^16 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..25

Cf. A002450 ((4^n-1)/3), A274906.

with(numtheory):
a:= proc(n) option remember; local k, t, d, h;
      for k from 1+ `if`(n=1, 0, ilog[4](a(n-1)*3+1))
      do t:= (4^k-1)/3;
         d:= max(factorset(t)[]);
         for h in [d+1, d-1] do
            if 2^ilog[2](h)=h then RETURN(t) fi
         od
      od
    end:
seq(a(n), n=1..17);  # Alois P. Heinz, Mar 04 2011

okQ[n_] := Module[{p = FactorInteger[n][[-1, 1]]}, IntegerQ[Log[2, p + 1]] || IntegerQ[Log[2, p - 1]]]; t = Table[(4^n-1)/3, {n,2,50}]; Select[t, okQ] (* T. D. Noe, Mar 04 2011 *)

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A274910 Largest prime factor of 11^n - 1.

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A366718 Largest prime factor of 12^n - 1.

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A358699 a(n) is the largest prime factor of 2^(prime(n) - 1) - 1.

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A187063 Numbers of the form (4^k - 1)/3 whose greatest prime divisor is of the form 2^q - 1 or 2^q + 1.

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