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%I #33 Apr 04 2021 01:00:51
%S 1,1,1,0,2,5,6,17,1,6,1,19,6,2,2,10,26
%N Minimal near-Wieferich A-value (absolute) for all primes in the interval [10^n, 10^(n+1)].
%C In other words, minimal value of abs(A) among A-values of all (n+1)-digit primes such that 2^{(p-1)/2} == +-1 + A*p (mod p^2).
%C a(n) = 0 indicates that at least one (n+1)-digit Wieferich prime (A001220) exists. In particular, a(3) = 0, because the interval [10^3, 10^4] contains the two Wieferich primes 1093 and 3511.
%C Clearly, the number of 0 terms is infinite if and only if A001220 is infinite.
%C a(9)-a(11) are from Crandall, Dilcher, Pomerance, 1997.
%C a(12) from data in Crandall, Dilcher, Pomerance, 1997 and Knauer, Richstein, 2005.
%C a(13)-a(14) from Knauer, Richstein, 2005.
%C In Crandall, Dilcher, Pomerance, 1997, a heuristic argument is given that predicts the number of Wieferich primes below some bound x to be about log(log(x)). If that heuristic is accurate, then one could expect the next 0 to occur at n with 9 <= n <= 24.
%H 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>, Mathematics of Computation 66 (1997), 433-449.
%H J. Knauer and J. Richstein, <a href="https://doi.org/10.1090/S0025-5718-05-01723-0">The continuing search for Wieferich primes</a>, Mathematics of Computation 74 (2005), 1559-1563.
%H PrimeGrid, <a href="https://www.primegrid.com/stats_ww.php">WW stats</a>
%H Sysadm@Nbg and PrimeGrid, <a href="https://web.archive.org/web/20170610212035/http://u-g-f.de:80/PRPNet/findlist_adv.php?proj=WFS">PRPNet findlist for project WFS</a> [Archived version at the Wayback Machine].
%e For n = 1: The A-values for the primes in the interval [10^1, 10^2] are 3, 5, -2, 8, -3, 14, 3, 18, -9, 9, 22, -18, -4, -18, -5, 1, 28, -30, -24, 3, -20. The smallest of these, by absolute value, is 1, so a(1) = 1.
%o (PARI) a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n)
%o a(n) = my(minm=nextprime(10^n)); forprime(p=10^n, 10^(n+1), if(p!=2, if(a258367(p) < minm, minm=a258367(p)))); minm
%Y Cf. A001220, A258367.
%K nonn,hard,more
%O 0,5
%A _Felix Fröhlich_, Mar 15 2019
%E a(15)-a(16) from _Felix Fröhlich_, Apr 03 2021
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