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.

5, 17, 8, 449, 26, 7, 557, 226, 18, 18, 19601, 1207, 1207, 148, 3, 132857, 54568, 1451, 606, 239, 19, 4486949, 2006776, 13543, 13543, 3469, 249, 38, 126664001, 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

Offset

1,1

Links

Table of n, a(n) for n=1..55.

Example

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.
Array starts:
n=1: 5, 17, 449, 557, 19601, 132857
n=2: 8, 26, 226, 1207, 54568, 2006776
n=3: 7, 18, 1207, 1451, 13543, 296449
n=4: 18, 148, 606, 13543, 296449, 17134811
n=5: 3, 239, 3469, 24675, 3414284, 36763941
n=6: 19, 249, 653, 39016, 14380864, 34998229

Prog

(PARI) primevec(initialp, vecsize) = my(v=[initialp]); while(#v < vecsize, v=concat(v, nextprime(v[#v]+1))); v
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)
array(rows, cols) = for(s=1, rows, for(t=1, cols, print1(a(s, t), ", ")); print(""))
array(5, 6) \\ print 5 X 6 array

Crossrefs

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).

Cf. A256236 (row n=1), A258787.

Sequence in context: A125636 A355658 A156323 * A276831 A180024 A356403

Adjacent sequences: A286813 A286814 A286815 * A286817 A286818 A286819

Keyword

nonn,tabl

Author

Felix Fröhlich, May 27 2017

Extensions

More terms from Max Alekseyev, Oct 10 2023

Status

approved