# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a250206 Showing 1-1 of 1 %I A250206 #69 Mar 13 2015 05:02:08 %S A250206 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 A250206 80,19,14,107,115,63,118,65,18,53,18,69,19,7,51,19,19,3,26,63,53,17, %U A250206 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 A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2). %C A250206 a(n) = least base b > 1 such that n is a Wieferich number (see A077816). %C A250206 At least, b = n^2+1 can satisfy this equation, so a(n) is defined for all n. %C A250206 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 A250206 Does every natural number (>1) appear in this sequence? If yes, do they appear infinitely many times? %C A250206 For prime n, a(n) = A185103(n), does there exist any composite n such that a(n) = A185103(n)? %H A250206 Eric Chen, Table of n, a(n) for n = 1..1000 %F A250206 a(prime(n)) = A039678(n) = A185103(prime(n)). %F A250206 a(A077816(n)) = 2. %F A250206 a(A242958(n)) <= 3. %e A250206 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 A250206 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 A250206 (PARI) a(n)=for(k=2,2^24,if((k^eulerphi(n))%(n^2)==1, return(k))) %Y A250206 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 A250206 nonn %O A250206 1,1 %A A250206 _Eric Chen_, Feb 21 2015 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE