A270776 Smallest non-Wieferich prime to base n, i.e., smallest prime p such that n^(p-1) != 1 (mod p^2).

2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2

Offset: 2

Author

Felix Fröhlich, Mar 22 2016

Keywords

• nonn

Comments

A256236 gives the smallest i such that a(i) = A000040(n).

a(n) > 2 iff A039951(n) = 2.

a(n) > 3 iff A268352(n) = 3.

Does every prime appear in the sequence?

It is easy to see that the answer to the previous question is "yes" if and only if A256236 is infinite.

The ABC-(k, Epsilon) conjecture with k >= 2 and Epsilon > 0 such that 1/(1/Epsilon + 1) + 1/k <= log(2)/(24*log(a)) implies that a(n) exists for all n (cf. Broughan, 2006; theorem 5.6).

Examples

			The sequence of base-17 Wieferich primes (A128668) starts 2, 3, 46021. Thus the smallest non-Wieferich prime to base 17 is 5 and hence a(17) = 5.
		

Links

• Felix Fröhlich, Table of n, a(n) for n = 2..10000
• K. A. Broughan, Relaxations of the abc conjecture using integer k'th roots, New Zealand Journal of Mathematics, 35 (2006), 121-136.

Crossrefs

Cf. A039951, A256236, A268352.

Programs

Mathematica

A270776[n_] := NestWhile[#+1 &, 2, CompositeQ[#] || PowerMod[n, #-1, #^2] == 1 &];
Array[A270776, 100, 2] (* Paolo Xausa, Aug 15 2025 *)

PARI

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