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 1-2 of 2 results.

A376253 Composite numbers k such that 2^(2^(k-1)-1) == 1 (mod k^2).

Original entry on oeis.org

4681, 15841, 42799, 52633, 220729, 647089, 951481, 1082401, 1145257, 1969417, 2215441, 3567481, 4835209, 5049001, 5681809, 6140161, 6334351, 8725753, 10712857, 11777599, 12327121, 13500313, 14709241, 22564081, 22849481, 22953673, 23828017, 27271151, 28758601, 30576151
Offset: 1

Views

Author

Thomas Ordowski, Sep 17 2024

Keywords

Comments

If 2^(k-1) == 1 (mod k) and 2^(2^(k-1)-1) == 1 (mod k), then 2^(2^(k-1)-1) == 1 (mod k^2). In fact, all such pseudoprimes are strong pseudoprimes to base 2.
Other terms; 951481 = 271*3511, 2215441 = 631*3511, 28758601 = 8191*3511, ... are not Fermat pseudoprimes to base 2, where 3511 is a Wieferich prime. The Wieferich prime 1093 cannot be a factor of these numbers (see A374953).

Crossrefs

Cf. A001220, A001262, A001567, A188465, A374841, A374953.

Programs

  • Mathematica
    q[k_] := Module[{m = MultiplicativeOrder[2, k^2]}, PowerMod[2, k-1, m] == 1]; Select[Range[1, 10^6, 2], CompositeQ[#] && q[#] &] (* Amiram Eldar, Sep 17 2024 *)
  • PARI
    is(k) = (k > 1) && k % 2 && !isprime(k) && Mod(2, znorder(Mod(2, k^2)))^(k-1) == 1; \\ Amiram Eldar, Sep 17 2024

Extensions

More terms from Amiram Eldar, Sep 17 2024

A376473 Numbers k such that 2^(2^(k-1)-1) == 1 (mod k^2) and 2^(k-1) =/= 1 (mod k).

Original entry on oeis.org

951481, 2215441, 28758601, 81844921, 1221936841, 10370479321, 16287076081, 26946809137, 33663998161, 35094800881, 134619011281, 188455112353, 299226038833, 314240366881, 383116075201, 594981050401, 1230227375833, 1572186445201, 2096189123113, 2377714473001
Offset: 1

Views

Author

Thomas Ordowski, Sep 24 2024

Keywords

Comments

The terms k of A374953 for which A002326((k-1)/2) is odd.
Numbers k in A376253 that are not strong pseudoprimes to base 2.
Every term of this sequence must have a Wieferich prime factor (for example, 951481 = 271 * 3511). The Wieferich prime 1093 cannot divide such a number (see A374953).

Crossrefs

Subsequence of A374953.
Cf. A001220, A002326, A001262, A001567, A376253.

Programs

  • Mathematica
    q[k_] := Module[{m = MultiplicativeOrder[2, k^2]}, PowerMod[2, k - 1, m] == 1]; Select[Range[1, 2300000, 2], PowerMod[2, # - 1, #] != 1 && q[#] &] (* Amiram Eldar, Sep 24 2024 *)
  • PARI
    is(k) = (k > 1) && k % 2 && !isprime(k) && Mod(2, k)^(k-1) != 1 && Mod(2, znorder(Mod(2, k^2)))^(k-1) == 1; \\ Amiram Eldar, Sep 24 2024
  • PARI
    list(lim)=my(v=List()); if(lim>3<<64, warning("May miss multiples of Wieferich primes > 2^64.")); forstep(n=10533,lim,7022, if(Mod(2, znorder(Mod(2, n^2)))^(n-1) == 1 && Mod(2,n)^n != 2, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Sep 24 2024

Extensions

More terms from Amiram Eldar, Sep 24 2024
Showing 1-2 of 2 results.

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

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