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

%I A270776 #23 Aug 16 2025 22:01:15
%S A270776 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,
%T A270776 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,
%U A270776 2,2,2,5,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2
%N A270776 Smallest non-Wieferich prime to base n, i.e., smallest prime p such that n^(p-1) != 1 (mod p^2).
%C A270776 A256236 gives the smallest i such that a(i) = A000040(n).
%C A270776 a(n) > 2 iff A039951(n) = 2.
%C A270776 a(n) > 3 iff A268352(n) = 3.
%C A270776 Does every prime appear in the sequence?
%C A270776 It is easy to see that the answer to the previous question is "yes" if and only if A256236 is infinite.
%C A270776 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).
%H A270776 Felix Fröhlich, <a href="/A270776/b270776.txt">Table of n, a(n) for n = 2..10000</a>
%H A270776 K. A. Broughan, <a href="http://nzjm.math.auckland.ac.nz/images/d/d6/Relaxations_of_the_ABC_Conjecture_using_integer_k%27th_roots.pdf">Relaxations of the abc conjecture using integer k'th roots</a>, New Zealand Journal of Mathematics, 35 (2006), 121-136.
%e A270776 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.
%t A270776 A270776[n_] := NestWhile[#+1 &, 2, CompositeQ[#] || PowerMod[n, #-1, #^2] == 1 &];
%t A270776 Array[A270776, 100, 2] (* _Paolo Xausa_, Aug 15 2025 *)
%o A270776 (PARI) a(n) = forprime(p=1, , if(Mod(n, p^2)^(p-1)!=1, return(p)))
%Y A270776 Cf. A039951, A256236, A268352.
%K A270776 nonn
%O A270776 2,1
%A A270776 _Felix Fröhlich_, Mar 22 2016