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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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