

A182297


Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q1)/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
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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^21)/2) = A002326((p1)/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), 13331336.


FORMULA

Odd numbers q such that A002326((q^21)/2) < q * A002326((q1)/2). Other positive odd integers satisfy the equality.  Thomas Ordowski, Feb 03 2014
Odd numbers q such that gcd(A165781((q1)/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 (2^61)/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(n1)) 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 * A020220 A251122 A084856
Adjacent sequences: A182294 A182295 A182296 * A182298 A182299 A182300


KEYWORD

nonn


AUTHOR

Felix FrÃ¶hlich, Apr 23 2012


STATUS

approved



