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Smallest prime dividing 2^n - 1.
19

%I #65 Mar 22 2024 19:38:14

%S 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,

%T 7,3,233,3,2147483647,3,7,3,31,3,223,3,7,3,13367,3,431,3,7,3,2351,3,

%U 127,3,7,3,6361,3,23,3,7,3,179951,3,2305843009213693951,3,7,3,31

%N Smallest prime dividing 2^n - 1.

%C 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

%C 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

%C 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

%H Max Alekseyev, <a href="/A049479/b049479.txt">Table of n, a(n) for n = 2..1236</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MersenneNumber.html">Mersenne Number</a>.

%F 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

%F For k >= 1, a(2k) = 3, a(6k-3)=7. - _Zak Seidov_, Mar 21 2014

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

%t 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 *)

%t FactorInteger[#][[1,1]]&/@(2^Range[2,70]-1) (* _Harvey P. Dale_, Nov 18 2019 *)

%o (Python)

%o from sympy import factorint

%o def A049479(n):

%o return min(factorint(2**n-1)) # _Chai Wah Wu_, Jun 03 2019

%Y Cf. A005420.

%K nonn

%O 2,1

%A _Olivier GĂ©rard_

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

%E Terms to a(500) in b-file from _T. D. Noe_, Dec 06 2006

%E a(501)-a(1060) in b-file from _Michel Marcus_, Sep 15 2017

%E a(1061)-a(1236) in b-file added at the suggestion of _Eric Chen_ by _Max Alekseyev_, Apr 25 2022