cp's OEIS Frontend

A003260 Largest prime factor of n-th Mersenne number (A001348(n)).

%I A003260 M2693 #62 Apr 20 2025 04:07:44
%S A003260 3,7,31,127,89,8191,131071,524287,178481,2089,2147483647,616318177,
%T A003260 164511353,2099863,13264529,20394401,3203431780337,2305843009213693951
%N A003260 Largest prime factor of n-th Mersenne number (A001348(n)).
%D A003260 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 A003260 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003260 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 A003260 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 A003260 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne primes</a>
%H A003260 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 A003260 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 A003260 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>
%F A003260 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 A003260 a[n_] := FactorInteger[ 2^Prime[n] - 1 ][[-1, 1]]; Table[ a[n], {n, 1, 18}] (* _Jean-François Alcover_, Dec 20 2011 *)
%o A003260 (PARI) a(n)=my(f=factor(2^prime(n)-1)[,1]);f[#f] \\ _Charles R Greathouse IV_, Dec 05 2012
%Y A003260 Cf. A000668 (a subsequence), A001348, A016047, A046800.
%K A003260 nonn,nice
%O A003260 1,1
%A A003260 _N. J. A. Sloane_