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A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A072708 A102478 A221048 * A270667 A020220 A251122

Adjacent sequences:  A182294 A182295 A182296 * A182298 A182299 A182300

KEYWORD

nonn

AUTHOR

Felix Fröhlich, Apr 23 2012

STATUS

approved

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Last modified December 10 23:13 EST 2016. Contains 279021 sequences.