cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

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

Keywords

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

Crossrefs

The even terms of A015919.
Cf. A295740.

Programs

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

A015919 Positive integers k such that 2^k == 2 (mod k).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 341, 347, 349, 353, 359, 367
Offset: 1

Views

Author

Robert G. Wilson v

Keywords

Comments

Includes 341 which is first pseudoprime to base 2 and distinguishes sequence from A008578.
First composite even term is a(14868) = 161038 = A006935(2). - Max Alekseyev, Feb 11 2015
If k is a term, then so is 2^k - 1. - Max Alekseyev, Sep 22 2016
Terms of the form 2^k - 2 correspond to k in A296104. - Max Alekseyev, Dec 04 2017
If 2^k - 1 is a term, then so is k. - Thomas Ordowski, Apr 27 2018

Links

Crossrefs

Contains A002997 as a subsequence.
The odd terms form A176997.
Cf. A000040, A001567, A008578.

Programs

  • Mathematica
    Prepend[ Select[ Range@370, PowerMod[2, #, #] == 2 &], {1, 2}] // Flatten (* Robert G. Wilson v, May 16 2018 *)
  • PARI
    is(n)=Mod(2,n)^n==2 \\ Charles R Greathouse IV, Mar 11 2014
  • Python
    def ok(n): return pow(2, n, n) == 2%n
print([k for k in range(1, 400) if ok(k)]) # Michael S. Branicky, Jun 03 2022

Formula

Equals {1} U A000040 U A001567 U A006935 = A001567 U A006935 U A008578. - Ray Chandler, Dec 07 2003; corrected by Max Alekseyev, Feb 11 2015

A296370 Numbers m such that 2^m == 3/2 (mod m).

Original entry on oeis.org

1, 111481, 465793, 79036177, 1781269903307, 250369632905747, 708229497085909, 15673900819204067
Offset: 1

Views

Author

Max Alekseyev, Dec 11 2017

Keywords

Comments

Equivalently, 2^(m+1) == 3 (mod m).
Also, numbers m such that 2^(m+1) - 2 is a Fermat pseudoprime base 2, i.e., 2^(m+1) - 2 belongs to A015919 and A006935.
Some larger terms (may be not in order): 2338990834231272653581, 341569682872976768698011746141903924998969680637.

Links

Crossrefs

Solutions to 2^m == k (mod m): this sequence (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Programs

  • Mathematica
    Select[Range[10^6], Divisible[2^(# + 1) - 3, #] &] (* Robert Price, Oct 11 2018 *)

Formula

a(n) = A296104(n) - 1.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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