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A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.

%I A246568 #57 Apr 29 2022 17:29:13
%S A246568 3,5,7,11,13,17,19,23,31,41,43,59,67,71,89,127,251,379,569,571,1093,
%T A246568 1427,1451,1733,2633,2659,2903,3511,13463,15329,15823,26107,60631,
%U A246568 546097,2549177,110057537,165322639,209227901,671499313,867457663,3520624567
%N A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.
%C A246568 The data section gives all terms up to 10^10. There are eight more terms up to 3*10^15 (see b-file).
%C A246568 A is essentially (A007663(n) modulo A000040(n))/2 (see Crandall et al. (1997), p. 437). The choice of the bound for A is rather arbitrary and selecting a larger A will result in more terms in a specific interval. For any p there exist two values of A whose sum is p, except when p is in A001220, in which case A = 0.
%H A246568 Jeppe Stig Nielsen, <a href="/A246568/b246568.txt">Table of n, a(n) for n = 1..50</a> (terms n= 1..49 by Felix Fröhlich)
%H A246568 R. Crandall, K. Dilcher and C. Pomerance, <a href="https://doi.org/10.1090/S0025-5718-97-00791-6">A search for Wieferich and Wilson primes</a>, Math. Comp. Vol. 66, Num. 217 (1997), 433-449.
%H A246568 J. Knauer and J. Richstein, <a href="https://doi.org/10.1090/S0025-5718-05-01723-0">The continuing search for Wieferich primes</a>, Math. Comp. Vol. 74, Num. 251 (2005), 1559-1563.
%H A246568 PrimeGrid, <a href="https://web.archive.org/web/20201126000718/https://www.primegrid.com/download/wieferich_list.pdf">Wieferich & near Wieferich Primes p < 11e15</a>
%H A246568 PrimeGrid, <a href="https://www.primegrid.com/stats_ww.php">WW Statistics</a>
%o A246568 (PARI) a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after _Charles R Greathouse IV_ in A258367
%o A246568 forprime(p=3, , if(a258367(p) < 10, print1(p, ", "))) \\ _Felix Fröhlich_, Apr 26 2022
%Y A246568 Cf. A001220, A195988, A241014, A244801.
%K A246568 nonn,hard
%O A246568 1,1
%A A246568 _Felix Fröhlich_, Aug 30 2014