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

3, 7, 31, 127, 89, 8191, 131071, 524287, 178481, 2089, 2147483647, 616318177, 164511353, 2099863, 13264529, 20394401, 3203431780337, 2305843009213693951

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Michel Marcus and Gord Palameta, Table of n, a(n) for n = 1..197 (first 95 terms from T. D. Noe, derived from Brillhart et al.)
• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• C. K. Caldwell, Mersenne primes
• P. Erdős and T. N. Shorey, On the greatest prime factor of 2^p-1 for a prime p and other expressions, Acta Arith. 30:3 (1976), pp. 257-265.
• C. L. Stewart, The greatest prime factor of a^n - b^n, Acta Arith. 26 (1975), pp. 427-433.
• S. S. Wagstaff, Jr., The Cunningham Project

Crossrefs

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

Programs

Mathematica

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

PARI

a(n)=my(f=factor(2^prime(n)-1)[,1]);f[#f] \\ Charles R Greathouse IV, Dec 05 2012

Formula

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