%I #69 Mar 13 2015 05:02:08

%S 2,5,8,7,7,17,18,15,26,7,3,17,19,19,26,31,38,53,28,7,19,3,28,17,57,19,

%T 80,19,14,107,115,63,118,65,18,53,18,69,19,7,51,19,19,3,26,63,53,17,

%U 18,57,134,19,338,161,3,31,28,41,53,107,264,115,19,127,99,161,143,65,28,99,11,55

%N Least base b > 1 such that b^A000010(n) = 1 (mod n^2).

%C a(n) = least base b > 1 such that n is a Wieferich number (see A077816).

%C At least, b = n^2+1 can satisfy this equation, so a(n) is defined for all n.

%C Least Wieferich number (>1) to base n: 2, 1093, 11, 1093, 2, 66161, 4, 3, 2, 3, 71, 2693, 2, 29, 4, 1093, 2, 5, 3, 281, 2, 13, 4, 5, 2, ...; each is a prime or 4. It is 4 if and only if n mod 72 is in the set {7, 15, 23, 31, 39, 47, 63}.

%C Does every natural number (>1) appear in this sequence? If yes, do they appear infinitely many times?

%C For prime n, a(n) = A185103(n), does there exist any composite n such that a(n) = A185103(n)?

%H Eric Chen, <a href="/A250206/b250206.txt">Table of n, a(n) for n = 1..1000</a>

%F a(prime(n)) = A039678(n) = A185103(prime(n)).

%F a(A077816(n)) = 2.

%F a(A242958(n)) <= 3.

%e a(30) = 107 since A000010(30) = 8, 30^2 = 900, and 107 is the least base b > 1 such that b^8 = 1 (mod 900).

%t f[n_] := Block[{b = 2, m = EulerPhi[n]}, While[ PowerMod[b, m, n^2] != 1, b++]; b]; f[1] = 2; Array[f, 72] (* _Robert G. Wilson v_, Feb 28 2015 *)

%o (PARI) a(n)=for(k=2,2^24,if((k^eulerphi(n))%(n^2)==1, return(k)))

%Y Cf. A039678, A185103, A125636, A039951, A247154, A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A077816, A242958, A242959, A241978, A242960, A241977, A253016, A245529.

%K nonn

%O 1,1

%A _Eric Chen_, Feb 21 2015