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A077816
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Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2).
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24
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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
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1,1
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COMMENTS
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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
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LINKS
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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.
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EXAMPLE
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A077815(3279) = 2^A000010(3279) mod 3279^2 = 2^2184 mod 10751841 = 1, therefore 3279 is a term.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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For another definition of Wieferich numbers, see A182297.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jun 18 2005
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STATUS
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approved
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