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

Author

Felix Fröhlich, Apr 23 2012

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

For another definition of Wieferich numbers, see A077816.

Cf. A002326.

Programs

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 *)

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

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