The On-Line Encyclopedia of Integer Sequences (OEIS)

A005420 revision #94

A005420

Largest prime factor of 2^n - 1.
(Formerly M2609)

3, 7, 5, 31, 7, 127, 17, 73, 31, 89, 13, 8191, 127, 151, 257, 131071, 73, 524287, 41, 337, 683, 178481, 241, 1801, 8191, 262657, 127, 2089, 331, 2147483647, 65537, 599479, 131071, 122921, 109, 616318177, 524287, 121369, 61681, 164511353, 5419

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

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

T. D. Noe, Charles R Greathouse IV and Amiram Eldar, Table of n, a(n) for n = 2..1206 (terms up to 500 from T. D. Noe, terms 501..1000 from Charles R Greathouse IV, terms 1001..1206 from Amiram Eldar)

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

factordb.com, Status of 2^1207-1. 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.

R. K. Guy, Letter to G. B. Huff & N. J. A. Sloane, Aug 1974

S. S. Wagstaff, Jr., The Cunningham Project

Eric Weisstein's World of Mathematics, Mersenne Number

FORMULA

a(n) = a(2n) iff a(n) > A002587(n). See A337431. - Thomas Ordowski, Jan 07 2014

A002326((a(n)-1)/2) = n iff n is odd or n is even such that a(n/2) != a(n). - Thomas Ordowski, Jan 11 2014

a(n) = A006530(A000225(n)). - Vincenzo Librandi, Jul 13 2016

a(n) = 2^n-1 = A000225(n) iff n is a Mersenne exponent (A000043). - Bernard Schott, Dec 11 2022

EXAMPLE

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

MATHEMATICA

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 *)

Table[FactorInteger[2^n - 1][[-1, 1]], {n, 2, 40}] (* Vincenzo Librandi, Jul 13 2016 *)

PROG

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

(PARI) a(n) = vecmax(factor(2^n-1)[, 1]); \\ Michel Marcus, Dec 15 2022

(Magma) [Maximum(PrimeDivisors(2^n-1)): n in [2..45]]; // Vincenzo Librandi, Jul 13 2016

CROSSREFS

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

Cf. similar sequences listed in A274906.

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

Sequence in context: A342660 A186522 A048857 * A212953 A161818 A161509

Adjacent sequences: A005417 A005418 A005419 * A005421 A005422 A005423

KEYWORD