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A077816 Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2).

%I A077816 #62 Mar 24 2023 15:29:27
%S A077816 1093,3279,3511,7651,10533,14209,17555,22953,31599,42627,45643,52665,
%T A077816 68859,94797,99463,127881,136929,157995,228215,298389,410787,473985,
%U A077816 684645,895167,1232361,2053935,2685501,3697083,3837523,6161805,11512569
%N A077816 Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2).
%C A077816 A077815(a(n)) = 1.
%C A077816 The only known primes are a(1)=A001220(1)=1093 and a(3)=A001220(2)=3511, the Wieferich primes.
%C A077816 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).
%C A077816 a(105)=A001220(3) in the sense that either both numbers are well-defined and equal, or else neither number exists. - _Jeppe Stig Nielsen_, Oct 16 2016
%H A077816 Max Alekseyev, <a href="/A077816/b077816.txt">Table of n, a(n) for n = 1..104</a> (all currently known terms)
%H A077816 T. Agoh, K. Dilcher, and L. Skula, <a href="http://dx.doi.org/10.1006/jnth.1997.2162">Fermat Quotients for Composite Moduli</a>, Journal of Number Theory 66(1), 1997, 29-50. doi: 10.1006/jnth.1997.2162
%H A077816 William D. Banks, Florian Luca, and Igor E. Shparlinski, <a href="http://www.math.missouri.edu/~bbanks/papers/2007_Wieferich_numbers.pdf">Estimates for Wieferich Numbers</a>, The Ramanujan Journal, December 2007, Volume 14, Issue 3, pp 361-378.
%H A077816 R. Crandall, K. Dilcher, and C. Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper111.pdf">A search for Wieferich and Wilson primes</a>, Mathematics of Computation, Volume 66, 1997.
%H A077816 R. Crandall and C. Pomerance, <a href="http://dx.doi.org/10.1007/0-387-28979-8">Prime Numbers: A Computational Perspective</a>, Springer, NY, 2001, p. 28.
%H A077816 Jiří Klaška, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Klaska/klaska2.html">A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers</a>, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.
%H A077816 Jiří Klaška, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Klaska/klaska6.html">Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem</a>, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
%e A077816 A077815(3279) = 2^A000010(3279) mod 3279^2 = 2^2184 mod 10751841 = 1, therefore 3279 is a term.
%t A077816 Reap[For[k = 1, k <= 10^8, k++, If[PowerMod[2, EulerPhi[k], k^2] == 1, Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Nov 17 2021 *)
%o A077816 (PARI) for(n=2, 10^9, if(Mod(2, n^2)^(eulerphi(n))==1, print1(n, ", "))); \\ _Felix Fröhlich_, May 27 2014
%o A077816 (Magma) [n: n in [1..8*10^5] | 2^EulerPhi(n) mod n^2 eq 1]; // _Vincenzo Librandi_, Dec 05 2015
%Y A077816 For another definition of Wieferich numbers, see A182297.
%Y A077816 Cf. A001220.
%K A077816 nonn
%O A077816 1,1
%A A077816 _Reinhard Zumkeller_, Nov 17 2002
%E A077816 More terms from _Emeric Deutsch_, Mar 05 2005
%E A077816 More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jun 18 2005