A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime.

21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, 417, 429, 453, 465, 483, 489, 495, 497, 505, 507, 525, 543, 555, 579, 597, 605, 609, 615, 627, 633, 651, 655

Offset

1,1

Comments

The primes in this sequence are A001220, the Wieferich primes. - Charles R Greathouse IV, Feb 02 2014

Odd prime p is a Wieferich prime if and only if A002326((p^2-1)/2) = A002326((p-1)/2). See the sixth comment to A001220 and my formula below. - Thomas Ordowski, Feb 03 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Z. Franco and C. Pomerance, On a conjecture of Crandall concerning the qx + 1 problem, Math. Comp. Vol. 64, No. 211 (1995), 1333-1336.

Formula

Odd numbers q such that A002326((q^2-1)/2) < q * A002326((q-1)/2). Other positive odd integers satisfy the equality. - Thomas Ordowski, Feb 03 2014

Odd numbers q such that gcd(A165781((q-1)/2), q) > 1. - Thomas Ordowski, Feb 12 2014

Example

21 is in the sequence because the multiplicative order of 2 mod 21 is 6, and (2^6-1)/21 = 3, which is not coprime to 21.

Maple

with(numtheory):
a:= proc(n) option remember; local q;
      for q from 2 +`if`(n=1, 1, a(n-1)) by 2
        while igcd((2^order(2, q)-1)/q, q)=1 do od; q
    end:
seq (a(n), n=1..60);  # Alois P. Heinz, Apr 23 2012

Mathematica

Select[Range[1, 799, 2], GCD[#, (2^MultiplicativeOrder[2, #] - 1)/#] > 1 &] (* Alonso del Arte, Apr 23 2012 *)

Prog

(PARI) is(n)=n%2 && gcd(lift(Mod(2, n^2)^znorder(Mod(2, n))-1)/n, n)>1 \\ Charles R Greathouse IV, Feb 02 2014

Crossrefs

For another definition of Wieferich numbers, see A077816.

Cf. A002326.

Sequence in context: A338330 A102478 A221048 * A338552 A339002 A339004

Adjacent sequences: A182294 A182295 A182296 * A182298 A182299 A182300

Keyword

nonn

Author

Felix Fröhlich, Apr 23 2012

Status

approved