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

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

Offset: 1

Author

N. J. A. Sloane, Richard C. Schroeppel

Keywords

• nonn
• nice

Comments

Of course, 2 is the only true prime here.

Numbers a(n)/2 form the odd terms of A130421. - Max Alekseyev, May 28 2014

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 (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

Numbers 2k such that 2^(2k-1) == 1 (mod k). - Thomas Ordowski, Nov 22 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

Terms of the form 2^k - 2 corresponds to k in A296104. - Max Alekseyev, Dec 04 2017

From Bernard Schott, Oct 11 2021: (Start)

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.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 91.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Max Alekseyev, Table of n, a(n) for n = 1..1319 (contains all terms below 2*10^15; first 156 terms from R. G. E. Pinch)
• N. G. W. H. Beeger, On even numbers m dividing 2^m-2, Amer. Math. Monthly, Vol. 58, No. 8 (1951), pp. 553-555.
• D. H. Lehmer, On the Converse of Fermat's Theorem II, Amer. Math. Monthly, Vol. 56, No. 5 (1949), pp. 300-309.
• A. Rotkiewicz and K. Ziemak, On Even Pseudoprimes, The Fibonacci Quarterly, Vol. 33, No. 2 (1995), pp. 123-125.
• Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
• Index entries for sequences related to pseudoprimes

Crossrefs

The even terms of A015919.

Cf. A295740.

Programs

Mathematica

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

PARI

is(n)=Mod(2,n)^n==2 && n%2==0 \\ Charles R Greathouse IV, Dec 02 2014

Extensions

More terms from Robert G. Wilson v

Corrected by T. D. Noe, May 27 2003

b-file corrected by Max Alekseyev, Oct 09 2016