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 A077816 Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2). 24
 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 (list; graph; refs; listen; history; text; internal format)
 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 well-defined 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, 29-50. doi: 10.1006/jnth.1997.2162 William D. Banks, Florian Luca, Igor E. Shparlinski, Estimates for Wieferich Numbers, The Ramanujan Journal, December 2007, Volume 14, Issue 3, pp 361-378. 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 * A291961 A001220 A265630 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 STATUS approved

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Last modified May 26 22:58 EDT 2020. Contains 334634 sequences. (Running on oeis4.)