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A244249 Table A(n,k) in which n-th row lists in increasing order all bases b to which p = prime(n) is a Wieferich prime (i.e., b^(p-1) is congruent to 1 mod p^2), read by antidiagonals.

%I A244249 #30 Oct 28 2021 06:29:50
%S A244249 5,9,8,13,10,7,17,17,18,18,21,19,24,19,3,25,26,26,30,9,19,29,28,32,31,
%T A244249 27,22,38,33,35,43,48,40,23,40,28,37,37,49,50,81,70,65,54,28,41,44,51,
%U A244249 67,94,80,75,62,42,14,45,46,57,68,112,89,110,68,63,41,115
%N A244249 Table A(n,k) in which n-th row lists in increasing order all bases b to which p = prime(n) is a Wieferich prime (i.e., b^(p-1) is congruent to 1 mod p^2), read by antidiagonals.
%H A244249 Alois P. Heinz, <a href="/A244249/b244249.txt">Rows n = 1..141, flattened</a>
%e A244249 Table starts with:
%e A244249 p =  2:   5,  9, 13, 17,  21,  25,  29,  33, ...
%e A244249 p =  3:   8, 10, 17, 19,  26,  28,  35,  37, ...
%e A244249 p =  5:   7, 18, 24, 26,  32,  43,  49,  51, ...
%e A244249 p =  7:  18, 19, 30, 31,  48,  50,  67,  68, ...
%e A244249 p = 11:   3,  9, 27, 40,  81,  94, 112, 118, ...
%e A244249 p = 13:  19, 22, 23, 70,  80,  89,  99, 146, ...
%e A244249 p = 17:  38, 40, 65, 75, 110, 131, 134, 155, ...
%p A244249 A:= proc(n, k) option remember; local p, b;
%p A244249       p:= ithprime(n);
%p A244249       for b from 1 +`if`(k=1, 1, A(n, k-1))
%p A244249         while b &^ (p-1) mod p^2<>1
%p A244249       do od; b
%p A244249     end:
%p A244249 seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # _Alois P. Heinz_, Jul 02 2014
%t A244249 A[n_, k_] := A[n, k] = Module[{p, b}, p = Prime[n]; For[b = 1 + If[k == 1, 1, A[n, k-1]], PowerMod[b, p-1, p^2] != 1, b++]; b]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 14}] // Flatten (* _Jean-François Alcover_, Mar 09 2015, after _Alois P. Heinz_ *)
%o A244249 (PARI) forprime(p=2, 10^1, print1("p=", p, ": "); for(a=2, 10^2, if(Mod(a, p^2)^(p-1)==1, print1(a, ", "))); print(""))
%Y A244249 Cf. A001220, A185103.
%Y A244249 First column of table is A039678.
%Y A244249 Main diagonal gives A280721.
%K A244249 nonn,tabl
%O A244249 1,1
%A A244249 _Felix Fröhlich_, Jun 23 2014