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A006935 Even pseudoprimes (or primes) to base 2: even n that divide 2^n - 2.
(Formerly M2190)
38

%I M2190 #144 Oct 11 2021 18:19:05

%S 2,161038,215326,2568226,3020626,7866046,9115426,49699666,143742226,

%T 161292286,196116194,209665666,213388066,293974066,336408382,

%U 377994926,410857426,665387746,667363522,672655726,760569694,1066079026,1105826338,1423998226,1451887438,1610063326,2001038066,2138882626,2952654706,3220041826

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

%C Of course, 2 is the only true prime here.

%C Numbers a(n)/2 form the odd terms of A130421. - _Max Alekseyev_, May 28 2014

%C 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 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 Numbers 2k such that 2^(2k-1) == 1 (mod k). - _Thomas Ordowski_, Nov 22 2016

%C 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 Terms of the form 2^k - 2 corresponds to k in A296104. - _Max Alekseyev_, Dec 04 2017

%C From _Bernard Schott_, Oct 11 2021: (Start)

%C Two significant dates in the history of these terms:

%C 1949: Derrick Henry Lehmer finds the smallest even pseudoprime to base 2, a(2) = 161038 (see Lehmer link).

%C 1951: Dutch mathematician N. G. W. H. Beeger proves that the number of even pseudoprimes is infinite (see Beeger link). (End)

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.

%D 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 R. K. Guy, Unsolved Problems in Number Theory, A12.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H 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 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 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 A. Rotkiewicz and K. Ziemak, <a href="http://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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPseudoprime.html">Fermat Pseudoprime</a>.

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%t Select[2*Range[5000000],PowerMod[2,#,#]==2&] (* _Harvey P. Dale_, Dec 02 2012 *)

%o (PARI) is(n)=Mod(2,n)^n==2 && n%2==0 \\ _Charles R Greathouse IV_, Dec 02 2014

%Y The even terms of A015919.

%Y Cf. A295740.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_, Richard C. Schroeppel

%E More terms from _Robert G. Wilson v_

%E Corrected by _T. D. Noe_, May 27 2003

%E b-file corrected by _Max Alekseyev_, Oct 09 2016

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