A252358 Smallest prime p such that (2^n)^(p-1) == 1 (mod p^2), i.e., smallest Wieferich prime to base 2^n.

2, 1093, 1093, 3, 1093, 5, 3, 7, 1093, 3, 5, 11, 3, 13, 7, 3, 1093, 17, 3, 19, 5, 3, 11, 23, 3, 5, 13, 3, 7, 29, 3, 31, 1093, 3, 17, 5, 3, 37, 19, 3, 5, 41, 3, 43, 11, 3, 23, 47, 3, 7, 5, 3, 13, 53, 3, 5, 7, 3, 29, 59, 3, 61, 31, 3, 1093, 5, 3, 67, 17, 3, 5

Offset

0,1

Comments

This sequence is bounded above by 1093, i.e., a(n) <= 1093 for any n, since any Wieferich prime to base 2 is also Wieferich to any base that is a power of two.

Do all primes <= 1093 appear in this sequence?

From Felix Fröhlich, Dec 10 2020: (Start)

The answer is yes. The smallest n such that a(n) = prime(n) is prime(n). The only exceptions are p = 2 and p = 1093 which first occur at n = 0 and n = 1, respectively.

Apparently, a(n) = 1093 if n is a power of 2 when n < 1093 and a(n) = 1093 if n is prime or a power of 2 when n > 1093. (End)

Links

Felix Fröhlich, Table of n, a(n) for n = 0..10000

Mathematica

Block[{k}, Table[k = 1; While[PowerMod[2, n (# - 1), #^2] != 1 &@ Prime@ k, k++]; Prime@ k, {n, 0, 70}] ] (* Michael De Vlieger, Dec 10 2020 *)

Prog

(PARI) a(n) = forprime(p=1, , if(Mod(2, p^2)^(n*(p-1))==1, return(p)))

Crossrefs

Cf. A001220, A039951.

Sequence in context: A039951 A247072 A282293 * A135618 A241921 A340290

Adjacent sequences: A252355 A252356 A252357 * A252359 A252360 A252361

Keyword

nonn

Author

Felix Fröhlich, Dec 22 2014

Status

approved