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 10^18 (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; updated by Max Alekseyev, Feb 16 2026
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:
1950: Derrick Henry Lehmer finds the smallest even pseudoprime to base 2, a(2) = 161038.
1951: Dutch mathematician N. G. W. H. Beeger proves that the number of even pseudoprimes is infinite (see Beeger link). (End)
From Amiram Eldar, Jan 28 2026: (Start)
Lehmer's discovery of a(2) was reported by Erdős (1950).
a(3), a(4) and a(9) were found by Beeger (1951).
a(15) and a(705) were found by S. Maciag (Sierpiński, 1953). (End)
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant 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., Vol. 22,, 2nd ed., AMS, 1988.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A12, pp. 44-50.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag, NY, 2004. See p. 91.
Wacław Sierpiński, Arytmetyka Teoretyczna, Warsaw, 1953.
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..7337 (all terms below 10^18; 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.
Xuyan Cheng, Paul Kinlaw, and Duc Nguyen, The Reciprocal Sum of Even Pseudoprimes, Journal of Integer Sequences, Vol. 29 (2026), Article 26.1.3.
Paul Erdős, On almost primes, Amer. Math. Monthly, Vol. 57, No. 6 (1950), pp. 404-407; alternative link.
Wayne L. McDaniel, Some pseudoprimes and related numbers having special forms, Mathematics of Computation, Vol. 53, No. 187 (1989), pp. 407-409.
Carl Pomerance and Samuel S. Wagstaff, Jr., Some thoughts on pseudoprimes, Bulletin (Académie serbe des sciences et des arts, Classe des sciences mathématiques et naturelles, Sciences mathématiques), Vol. 46 (2021), pp. 53-72; arXiv preprint, arXiv:2103.00679 [math.NT], 2021; JSTOR link; author's link.
A. Rotkiewicz and K. Ziemak, On Even Pseudoprimes, The Fibonacci Quarterly, Vol. 33, No. 2 (1995), pp. 123-125.
Wacław Sierpiński, Elementary Theory of Numbers, Państ. Wydaw. Nauk., Warsaw, 1964, pp. 214-216.
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
FORMULA
Sum_{n>=2} 1/a(n) is in the interval (0.000011, 0.0059) (Cheng et al., 2026). - Amiram Eldar, Jan 28 2026
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
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane and 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