This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A235366 #33 Apr 28 2022 10:27:46
%S A235366 13,5,11,7,1093,5,13,11,23,5,797161,547,11,5,1871,7,1597,5,13,23,47,5,
%T A235366 11,398581,13,5,59,7,683,5,13,103,11,5,13097927,1597,13,5,83,7,431,5,
%U A235366 11,47,1223,5,491,11,13,5,107,7,11,5,13,59,14425532687,5,603901,683,13,5,11,7,221101,5,13,11
%N A235366 Smallest odd prime factor of 3^n - 1.
%C A235366 Levi Ben Gerson (1288-1344) proved that 3^n - 1 = 2^m has no solution in integers if n > 2, by showing that 3^n - l has an odd prime factor. His proof uses remainders after division of powers of 3 by 8 and powers of 2 by 8; see the Lenstra and Peterson links. For an elegant short proof, see the Franklin link. - Sondow
%C A235366 One way to prove it is by the use of congruences. The powers of 3, modulo 80, are 3, 9, 27, 1, 3, 9, 27, 1, 3, 9, 27, 1, ... The powers of 2 are 2, 4, 8, 16, 32, 64, 48, 16, 32, 64, 48, 16, ... - _Alonso del Arte_, Jan 20 2014
%D A235366 L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.
%H A235366 Max Alekseyev, <a href="/A235366/b235366.txt">Table of n, a(n) for n = 3..796</a> (terms to a(660) from Charles R Greathouse IV)
%H A235366 P. Franklin, <a href="http://www.jstor.org/stable/2298495">Problem 2927</a>, Amer. Math. Monthly, 30 (1923), p. 81.
%H A235366 A. Herschfeld, <a href="http://dx.doi.org/10.1090/S0002-9904-1936-06275-0">The equation 2^x - 3^y = d</a>, Bull. Amer. Math. Soc., 42 (1936), 231-234.
%H A235366 H. Lenstra <a href="http://www.msri.org/publications/ln/msri/1998/mandm/lenstra/1/index.html">Harmonic Numbers</a>, MSRI, 1998.
%H A235366 J. J. O'Connor and E. F. Robertson, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Levi.html">Levi ben Gerson</a>, The MacTutor History of Mathematics archive, 2009.
%H A235366 I. Peterson, <a href="http://archive.is/iRXz">Medieval Harmony</a>, Math Trek, MAA, 2012.
%H A235366 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gersonides">Gersonides</a>
%F A235366 a(4n) = 5 as 3^(4n)-1 = (3^4)^n-1 = 81^n-1 = (80+1)^n-1 == 0 (mod 5).
%F A235366 a(6+12n) = 7 as 3^(6+12n)-1 = (3^6)^(1+2n)-1 = 729^(1+2n)-1 = (728+1)^(1+2n)-1 == 1^(1+2n)-1 == 0 (mod 7), but 729^(1+2n)-1 = (730-1)^(1+2n)-1 == (-1)^(1+2n)-1 == -2 (mod 5).
%e A235366 3^3 - 1 = 26 = 2 * 13, so a(3) = 13.
%e A235366 3^4 - 1 = 80 = 2^4 * 5, so a(4) = 5.
%e A235366 3^5 - 1 = 242 = 2 * 11^2, so a(5) = 11.
%t A235366 Table[FactorInteger[3^n - 1][[2, 1]], {n, 3, 50}]
%o A235366 (PARI) a(n)=factor(3^n>>valuation(3^n-1,2))[1,1] \\ _Charles R Greathouse IV_, Jan 20 2014
%Y A235366 See A235365 for 3^n + 1.
%Y A235366 Cf. also A003586 (products 2^m * 3^n), A006899, A061987, A108906.
%K A235366 nonn
%O A235366 3,1
%A A235366 _Jonathan Sondow_, Jan 19 2014