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

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

1,1

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.

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

Mathematica

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

Prog

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

Crossrefs

The even terms of A015919.

Cf. A295740.

Sequence in context: A167518 A178168 A271669 * A070833 A176584 A152475

Adjacent sequences: A006932 A006933 A006934 * A006936 A006937 A006938

Keyword

nonn,nice

Author

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

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

Status

approved