A250206 Least base b > 1 such that b^A000010(n) = 1 (mod n^2).

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

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: A057929 A353260 A154127 * A138371 A140053 A346180

Adjacent sequences: A250203 A250204 A250205 * A250207 A250208 A250209

Keyword

nonn

Author

Eric Chen, Feb 21 2015

Status

approved