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A174422 1st Wieferich prime base prime(n). 11
1093, 11, 2, 5, 71, 2, 2, 3, 13, 2, 7, 2, 2, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Smallest prime p such that p^2 divides prime(n)^(p-1) - 1.

Smallest prime p such that p divides the Fermat quotient q_p((prime(n)) = (prime(n)^(p-1) - 1)/p.

See additional comments, links, and cross-refs in A039951.

a(15) = A039951(47) > 4.1*10^13.

LINKS

Table of n, a(n) for n=1..14.

Wikipedia, Generalized Wieferich primes.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (or 0 if unknown)

FORMULA

a(n) = A039951(prime(n)).

a(n) = 2 if and only if prime(n) == 1 (mod 4). [Jonathan Sondow, Aug 29 2010]

EXAMPLE

a(1) = 1093 is the first Wieferich prime A001220. a(2) = 11 is the first Mirimanoff prime A014127.

MATHEMATICA

f[n_] := Block[{b = Prime@ n, p = 2}, While[ PowerMod[b, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 14]

PROG

(PARI) forprime(a=2, 20, forprime(p=2, 10^9, if(Mod(a, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ Felix Fröhlich, Jun 27 2014

CROSSREFS

Cf. A001220, A014127, A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A125636 = smallest prime p such that prime(n)^2 divides p^(prime(n)-1) - 1.

Cf. A178871 = 2nd Wieferich prime base prime(n).

Sequence in context: A281001 A271100 A258368 * A255838 A253234 A138698

Adjacent sequences:  A174419 A174420 A174421 * A174423 A174424 A174425

KEYWORD

hard,more,nonn

AUTHOR

Jonathan Sondow, Mar 19 2010

STATUS

approved

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Last modified January 25 19:09 EST 2020. Contains 331249 sequences. (Running on oeis4.)