

A006935


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


38



2, 161038, 215326, 2568226, 3020626, 7866046, 9115426, 49699666, 143742226, 161292286, 196116194, 209665666, 213388066, 293974066, 336408382, 377994926, 410857426, 665387746, 667363522, 672655726, 760569694, 1066079026, 1105826338, 1423998226, 1451887438, 1610063326, 2001038066, 2138882626, 2952654706, 3220041826
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OFFSET

1,1


COMMENTS

Of course, 2 is the only true prime here.
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
Corollary (RotkiewiczZiemak, 1995): 2(2^p1)(2^q1) 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
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
Two significant dates in the history of these terms:
1949: Derrick Henry Lehmer finds the smallest even pseudoprime to base 2, a(2) = 161038 (see Lehmer link).
1951: Dutch mathematician N. G. W. H. Beeger proves that the number of even pseudoprimes is infinite (see Beeger link). (End)


REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
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.
R. K. Guy, Unsolved Problems in Number Theory, A12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

A. Rotkiewicz and K. Ziemak, On Even Pseudoprimes, The Fibonacci Quarterly, Vol. 33, No. 2 (1995), pp. 123125.


MATHEMATICA

Select[2*Range[5000000], PowerMod[2, #, #]==2&] (* Harvey P. Dale, Dec 02 2012 *)


PROG



CROSSREFS



KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



