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 A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2). 1
 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, 80, 19, 14, 107, 115, 63, 118, 65, 18, 53, 18, 69, 19, 7, 51, 19, 19, 3, 26, 63, 53, 17, 18, 57, 134, 19, 338, 161, 3, 31, 28, 41, 53, 107, 264, 115, 19, 127, 99, 161, 143, 65, 28, 99, 11, 55 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(n) = least base b > 1 such that n is a Wieferich number (see A077816). At least, b = n^2+1 can satisfy this equation, so a(n) is defined for all n. 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}. Does every natural number (>1) appear in this sequence? If yes, do they appear infinitely many times? For prime n, a(n) = A185103(n), does there exist any composite n such that a(n) = A185103(n)? LINKS Eric Chen, Table of n, a(n) for n = 1..1000 FORMULA a(prime(n)) = A039678(n) = A185103(prime(n)). a(A077816(n)) = 2. a(A242958(n)) <= 3. EXAMPLE 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). MATHEMATICA 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 *) PROG (PARI) a(n)=for(k=2, 2^24, if((k^eulerphi(n))%(n^2)==1, return(k))) CROSSREFS 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. Sequence in context: A287013 A057929 A154127 * A138371 A140053 A103311 Adjacent sequences:  A250203 A250204 A250205 * A250207 A250208 A250209 KEYWORD nonn AUTHOR Eric Chen, Feb 21 2015 STATUS approved

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Last modified January 19 14:53 EST 2020. Contains 331049 sequences. (Running on oeis4.)