

A077816


Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2).


18



1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167, 1232361, 2053935, 2685501, 3697083, 3837523, 6161805, 11512569
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OFFSET

1,1


COMMENTS

A077815(a(n)) = 1.
The only known primes are a(1)=A001220(1)=1093 and a(3)=A001220(2)=3511, the Wieferich primes.
If there are finitely many Wieferich primes (A001220), this sequence is finite. In particular, unless there are other Wieferich primes besides 1093 and 3511, this sequence consists of 104 terms with the largest being 16547533489305 (Agoh et al., 1997).
a(105)=A001220(3) in the sense that either both numbers are welldefined and equal, or else neither number exists.  Jeppe Stig Nielsen, Oct 16 2016


LINKS

Max Alekseyev, Table of n, a(n) for n = 1..104 (all currently known terms)
T. Agoh, K. Dilcher, L. Skula, Fermat Quotients for Composite Moduli, Journal of Number Theory 66(1), 1997, 2950. doi: 10.1006/jnth.1997.2162
William D. Banks, Florian Luca, Igor E. Shparlinski, Estimates for Wieferich Numbers, 2007.
R. Crandall, K. Dilcher, C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Volume 66, 1997.
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001, p. 28.


EXAMPLE

A077815(3279) = 2^A000010(3279) mod 3279^2 = 2^2184 mod 10751841 = 1, therefore 3279 is a term.


PROG

(PARI) for(n=2, 10^9, if(Mod(2, n^2)^(eulerphi(n))==1, print1(n, ", "))); \\ Felix FrÃ¶hlich, May 27 2014
(MAGMA) [n: n in [1..8*10^5]  2^EulerPhi(n) mod n^2 eq 1]; // Vincenzo Librandi, Dec 05 2015


CROSSREFS

For another definition of Wieferich numbers, see A182297.
Cf. A001220.
Sequence in context: A023698 A038469 A246503 * A001220 A265630 A270833
Adjacent sequences: A077813 A077814 A077815 * A077817 A077818 A077819


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Nov 17 2002


EXTENSIONS

More terms from Emeric Deutsch, Mar 05 2005
More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jun 18 2005
bfile added by Max Alekseyev, Dec 18 2011


STATUS

approved



