%I M2609 #100 Dec 17 2022 20:01:59

%S 3,7,5,31,7,127,17,73,31,89,13,8191,127,151,257,131071,73,524287,41,

%T 337,683,178481,241,1801,8191,262657,127,2089,331,2147483647,65537,

%U 599479,131071,122921,109,616318177,524287,121369,61681,164511353,5419

%N Largest prime factor of 2^n - 1.

%D J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, Charles R Greathouse IV and Amiram Eldar, <a href="/A005420/b005420.txt">Table of n, a(n) for n = 2..1206</a> (terms up to 500 from T. D. Noe, terms 501..1000 from Charles R Greathouse IV, terms 1001..1206 from Amiram Eldar)

%H J. Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

%H factordb.com, <a href="http://factordb.com/index.php?query=2%5E1207-1">Status of 2^1207-1</a>. The factorization of the composite factor C337 of 2^1207-1 with 337 decimal digits is considered by many to be the most desired open factorization problem.

%H R. K. Guy, <a href="/A002968/a002968_1.pdf">Letter to G. B. Huff & N. J. A. Sloane, Aug 1974</a>

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MersenneNumber.html">Mersenne Number</a>

%F a(n) = a(2n) iff a(n) > A002587(n). See A337431. - _Thomas Ordowski_, Jan 07 2014

%F a(n) = A006530(A000225(n)). - _Vincenzo Librandi_, Jul 13 2016

%F a(n) = 2^n-1 = A000225(n) iff n is a Mersenne exponent (A000043). - _Bernard Schott_, Dec 11 2022

%e 2^6 - 1 = 63 = 3*21 = 9*7, so a(6) = 7.

%t a[n_] := a[n] = FactorInteger[2^n-1] // Last // First; Table[Print[{n, a[n]}, If[2^n-1 == a[n], " Mersenne prime", " "]]; a[n], {n, 2, 127}] (* _Jean-François Alcover_, Dec 11 2012 *)

%t Table[FactorInteger[2^n - 1][[-1, 1]], {n, 2, 40}] (* _Vincenzo Librandi_, Jul 13 2016 *)

%o (PARI) for(n=2,44, v=factor(2^n-1)[,1]; print1(v[#v]", "));

%o (PARI) a(n) = vecmax(factor(2^n-1)[,1]); \\ _Michel Marcus_, Dec 15 2022

%o (Magma) [Maximum(PrimeDivisors(2^n-1)): n in [2..45]]; // _Vincenzo Librandi_, Jul 13 2016

%Y Cf. A000043, A000225, A006530, A002326, A002587.

%Y Cf. similar sequences listed in A274906.

%Y Cf. A337431 (a(n)=a(2n)), A359063 (a(n)=a(2n)=a(4n)), A359088.

%K nonn

%O 2,1

%A _N. J. A. Sloane_

%E Description corrected by _Michael Somos_, Feb 24 2002

%E More terms from _Rick L. Shepherd_, Aug 22 2002

%E Incorrect comments removed by _Michel Marcus_, Dec 15 2022