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A286816 Smallest b such that the k consecutive primes starting with prime(n) are all base-b Wieferich primes, i.e., satisfy b^(p-1) == 1 (mod p^2). Square array A(n, k), read by antidiagonals downwards.

%I A286816 #33 Oct 12 2023 08:57:06
%S A286816 5,17,8,449,26,7,557,226,18,18,19601,1207,1207,148,3,132857,54568,
%T A286816 1451,606,239,19,4486949,2006776,13543,13543,3469,249,38,126664001,
%U A286816 20950343,296449,296449,24675,653,423,28,2363321449,230695118,23250274,17134811,3414284,39016,5649,28,28,5229752849,5229752849,882345432,741652533,36763941,14380864,217682,26645,63,14
%N A286816 Smallest b such that the k consecutive primes starting with prime(n) are all base-b Wieferich primes, i.e., satisfy b^(p-1) == 1 (mod p^2). Square array A(n, k), read by antidiagonals downwards.
%e A286816 The sequence of base-226 Wieferich primes starts 3, 5, 7, 97, 157, ... Since 226 is the smallest b such that the three consecutive primes starting with prime(2) = 3 are base-b Wieferich primes, A(2, 3) = 226.
%e A286816 Array starts:
%e A286816 n=1: 5, 17, 449, 557, 19601, 132857
%e A286816 n=2: 8, 26, 226, 1207, 54568, 2006776
%e A286816 n=3: 7, 18, 1207, 1451, 13543, 296449
%e A286816 n=4: 18, 148, 606, 13543, 296449, 17134811
%e A286816 n=5: 3, 239, 3469, 24675, 3414284, 36763941
%e A286816 n=6: 19, 249, 653, 39016, 14380864, 34998229
%o A286816 (PARI) primevec(initialp, vecsize) = my(v=[initialp]); while(#v < vecsize, v=concat(v, nextprime(v[#v]+1))); v
%o A286816 a(n, k) = my(v=primevec(prime(n), k), b=2, i=0); while(1, for(x=1, #v, if(Mod(b, v[x]^2)^(v[x]-1)!=1, i++; break)); if(i==0, return(b)); b++; i=0)
%o A286816 array(rows, cols) = for(s=1, rows, for(t=1, cols, print1(a(s, t), ", ")); print(""))
%o A286816 array(5, 6) \\ print 5 X 6 array
%Y A286816 Columns: A039678 (k=1), A259075 (k=2), A344827 (k=3), A344828 (k=4), A344829 (k=5), A344830 (k=6), A344831 (k=7), A344832 (k=8).
%Y A286816 Cf. A256236 (row n=1), A258787.
%K A286816 nonn,tabl
%O A286816 1,1
%A A286816 _Felix Fröhlich_, May 27 2017
%E A286816 More terms from _Max Alekseyev_, Oct 10 2023