A049479 Smallest prime dividing 2^n - 1.

3, 7, 3, 31, 3, 127, 3, 7, 3, 23, 3, 8191, 3, 7, 3, 131071, 3, 524287, 3, 7, 3, 47, 3, 31, 3, 7, 3, 233, 3, 2147483647, 3, 7, 3, 31, 3, 223, 3, 7, 3, 13367, 3, 431, 3, 7, 3, 2351, 3, 127, 3, 7, 3, 6361, 3, 23, 3, 7, 3, 179951, 3, 2305843009213693951, 3, 7, 3, 31

Offset: 2

Author

Olivier Gérard

Keywords

• nonn

Comments

If p is prime then a(p) == 1 (mod p). Are there composite numbers k such that a(k) == 1 (mod k)? - Thomas Ordowski, Jan 27 2014

Yes, up to 1200, the following composites have the desired property: 169, 221, 323, 611, 779, 793, 923, 1121, 1159. - Michel Marcus, Jan 28 2014

a(n) <= a(lpf(n)) for every n, where lpf(n) = A020639(n). For which n is a(n) < a(lpf(n))? See A236769. - Thomas Ordowski, Jan 30 2014

Examples

			a(6)=3 since 2^6 - 1 = 63 = 3^2*7.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 2..1236
• Eric Weisstein's World of Mathematics, Mersenne Number.

Crossrefs

Cf. A005420.

Programs

Mathematica

a = {}; Do[w = 2^n - 1; c = FactorInteger[w]; b = c[[1]][[1]]; AppendTo[a, b], {n, 2, 65}]; a (* Artur Jasinski, Dec 11 2007 *)
FactorInteger[#][[1,1]]&/@(2^Range[2,70]-1) (* Harvey P. Dale, Nov 18 2019 *)

Python

from sympy import factorint
def A049479(n):
    return min(factorint(2**n-1)) # Chai Wah Wu, Jun 03 2019

Formula

a(n) > lpf(n) while a(2k) = 3 and a(2k+1) > 2*lpf(2k+1), where lpf(m) = A020639(m). - Thomas Ordowski, Jan 29 2014

For k >= 1, a(2k) = 3, a(6k-3)=7. - Zak Seidov, Mar 21 2014

Extensions

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

Terms to a(500) in b-file from T. D. Noe, Dec 06 2006

a(501)-a(1060) in b-file from Michel Marcus, Sep 15 2017

a(1061)-a(1236) in b-file added at the suggestion of Eric Chen by Max Alekseyev, Apr 25 2022