[go: up one dir, main page]

login
A005420
Largest prime factor of 2^n - 1.
(Formerly M2609)
34
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
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.
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
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. similar sequences listed in A274906.
Cf. A337431 (a(n)=a(2n)), A359063 (a(n)=a(2n)=a(4n)), A359088.
Sequence in context: A342660 A186522 A048857 * A212953 A161818 A161509
KEYWORD
nonn
EXTENSIONS
Description corrected by Michael Somos, Feb 24 2002
More terms from Rick L. Shepherd, Aug 22 2002
Incorrect comments removed by Michel Marcus, Dec 15 2022
STATUS
approved