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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, and 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, and Igor E. Shparlinski, Estimates for Wieferich Numbers, The Ramanujan Journal, December 2007, Volume 14, Issue 3, pp 361-378.
R. Crandall, K. Dilcher, and 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.
Jiří Klaška, A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.
Jiří Klaška, Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
EXAMPLE
A077815(3279) = 2^A000010(3279) mod 3279^2 = 2^2184 mod 10751841 = 1, therefore 3279 is a term.
MATHEMATICA
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 *)
PROG
(PARI) for(n=2, 10^9, if(Mod(2, n^2)^(eulerphi(n))==1, print1(n, ", "))); \\ Felix Fröhlich, 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
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 March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)