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

2,1

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).

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.

Example

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.

Prog

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

Crossrefs

Cf. A039951, A256236, A268352.

Sequence in context: A156384 A306249 A064656 * A056608 A091787 A087040

Adjacent sequences: A270773 A270774 A270775 * A270777 A270778 A270779

Keyword

nonn

Author

Felix Fröhlich, Mar 22 2016

Status

approved