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

Author

Eric Chen, Feb 21 2015

Keywords

• nonn

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)?

Examples

			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).
		

Links

• Eric Chen, Table of n, a(n) for n = 1..1000

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.

Programs

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 *)

PARI

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

Formula

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

a(A077816(n)) = 2.

a(A242958(n)) <= 3.