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Largest prime factor of n-th Mersenne number (A001348(n)).
(Formerly M2693)
12

%I M2693 #58 Oct 29 2023 01:44:34

%S 3,7,31,127,89,8191,131071,524287,178481,2089,2147483647,616318177,

%T 164511353,2099863,13264529,20394401,3203431780337,2305843009213693951

%N Largest prime factor of n-th Mersenne number (A001348(n)).

%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 Michel Marcus and Gord Palameta, <a href="/A003260/b003260.txt">Table of n, a(n) for n = 1..197</a> (first 95 terms from T. D. Noe, derived from Brillhart et al.)

%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 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne primes</a>

%H P. Erdős and T. N. Shorey, <a href="http://www.renyi.hu/~p_erdos/1976-28.pdf">On the greatest prime factor of 2^p—1 for a prime p and other expressions</a>, Acta Arith. 30:3 (1976), pp. 257-265.

%H C. L. Stewart, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa26/aa2647.pdf">The greatest prime factor of a^n - b^n</a>, Acta Arith. 26 (1975), pp. 427-433.

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

%F Let p = prime(n). Erdős & Shorey show that a(n) >= kp log p for some effectively computable k >= 1. (Presumably k can be chosen as 7/log 27.) - _Charles R Greathouse IV_, Dec 05 2012

%t a[n_] := FactorInteger[ 2^Prime[n] - 1 ][[-1, 1]]; Table[ a[n], {n, 1, 18}] (* _Jean-François Alcover_, Dec 20 2011 *)

%o (PARI) a(n)=my(f=factor(2^prime(n)-1)[,1]);f[#f] \\ _Charles R Greathouse IV_, Dec 05 2012

%Y Cf. A000668 (a subsequence), A001348, A016047, A046800.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_