|
|
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.
|
|
LINKS
|
|
|
FORMULA
|
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).
|
|
KEYWORD
|
hard,more,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|