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A006935 Even pseudoprimes (or primes) to base 2: even n that divide 2^n - 2.

%I A006935 M2190 #150 Apr 05 2025 10:27:27
%S A006935 2,161038,215326,2568226,3020626,7866046,9115426,49699666,143742226,
%T A006935 161292286,196116194,209665666,213388066,293974066,336408382,
%U A006935 377994926,410857426,665387746,667363522,672655726,760569694,1066079026,1105826338,1423998226,1451887438,1610063326,2001038066,2138882626,2952654706,3220041826
%N A006935 Even pseudoprimes (or primes) to base 2: even n that divide 2^n - 2.
%C A006935 Of course, 2 is the only true prime here.
%C A006935 Numbers a(n)/2 form the odd terms of A130421. - _Max Alekseyev_, May 28 2014
%C A006935 a(n) == 2 (mod 4), hence there are no consecutive even numbers in this sequence. The closest two terms below 2*10^15 (as computed by Alekseyev) are a(2) = 161038 and a(3) = 215326 with a(3) - a(2) = 54288. Do smaller gaps exist? - _Charles R Greathouse IV_, Dec 02 2014
%C A006935 Corollary (Rotkiewicz-Ziemak, 1995): 2(2^p-1)(2^q-1) is a pseudoprime if and only if 2(2^(pq)-1) is a pseudoprime, where p,q are distinct primes. - _Thomas Ordowski_, Apr 09 2016
%C A006935 Numbers 2k such that 2^(2k-1) == 1 (mod k). - _Thomas Ordowski_, Nov 22 2016
%C A006935 There exist even pseudoprimes that are not squarefree, with the smallest being 190213279479817426 = 2 * 7 * 79 * 1951 * 3511^2 * 7151 (cf. A295740). - _Max Alekseyev_, Nov 26 2017
%C A006935 Terms of the form 2^k - 2 corresponds to k in A296104. - _Max Alekseyev_, Dec 04 2017
%C A006935 From _Bernard Schott_, Oct 11 2021: (Start)
%C A006935 Two significant dates in the history of these terms:
%C A006935 1949: Derrick Henry Lehmer finds the smallest even pseudoprime to base 2, a(2) = 161038 (see Lehmer link).
%C A006935 1951: Dutch mathematician N. G. W. H. Beeger proves that the number of even pseudoprimes is infinite (see Beeger link). (End)
%D A006935 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
%D A006935 J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr., Factorizations of b^n+/-1 b=2, 3, 5, 6, 7, 10, 11, 12 up to high powers, Contemporary Math. v.22.
%D A006935 R. K. Guy, Unsolved Problems in Number Theory, A12.
%D A006935 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 91.
%D A006935 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006935 Max Alekseyev, <a href="/A006935/b006935.txt">Table of n, a(n) for n = 1..1319</a> (contains all terms below 2*10^15; first 156 terms from R. G. E. Pinch)
%H A006935 N. G. W. H. Beeger, <a href="http://www.jstor.org/stable/2306320">On even numbers m dividing 2^m-2</a>, Amer. Math. Monthly, Vol. 58, No. 8 (1951), pp. 553-555.
%H A006935 D. H. Lehmer, <a href="https://www.jstor.org/stable/2306041">On the Converse of Fermat's Theorem II</a>, Amer. Math. Monthly, Vol. 56, No. 5 (1949), pp. 300-309.
%H A006935 A. Rotkiewicz and K. Ziemak, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-2/rotkiewicz.pdf">On Even Pseudoprimes</a>, The Fibonacci Quarterly, Vol. 33, No. 2 (1995), pp. 123-125.
%H A006935 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatPseudoprime.html">Fermat Pseudoprime</a>.
%H A006935 <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>
%t A006935 Select[2*Range[5000000],PowerMod[2,#,#]==2&] (* _Harvey P. Dale_, Dec 02 2012 *)
%o A006935 (PARI) is(n)=Mod(2,n)^n==2 && n%2==0 \\ _Charles R Greathouse IV_, Dec 02 2014
%Y A006935 The even terms of A015919.
%Y A006935 Cf. A295740.
%K A006935 nonn,nice
%O A006935 1,1
%A A006935 _N. J. A. Sloane_, Richard C. Schroeppel
%E A006935 More terms from _Robert G. Wilson v_
%E A006935 Corrected by _T. D. Noe_, May 27 2003
%E A006935 b-file corrected by _Max Alekseyev_, Oct 09 2016