A125636 Smallest odd prime base q such that p^2 divides q^(p-1) - 1, where p = prime(n).

5, 17, 7, 19, 3, 19, 131, 127, 263, 41, 229, 691, 313, 19, 53, 521, 53, 601, 1301, 11, 619, 31, 269, 3187, 53, 181, 43, 317, 499, 373, 911, 659, 19, 3659, 313, 751, 233, 4373, 3307, 419, 2591, 313, 1249, 2897, 349, 709, 331, 1973, 1933, 503, 821, 977, 2371, 263

Offset: 1

Author

Alexander Adamchuk, Nov 28 2006

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• W. Keller and J. Richstein, Fermat quotients that are divisible by p.

Crossrefs

Cf. A125637 (analogous with p^3 instead of p^2).

Cf. A125609 (q=3), A125610 (q=5), A125611 (q=7), A125612 (q=11), A125632 (q=13), A125633 (q=17), A125634 (q=19): sequences of smallest prime p such that q^n divides p^(q-1) - 1.

Programs

Maple

a:= proc(p)
  local q;
  q:= 3;
  while (q &^ (p-1) - 1) mod p^2 <> 0 do
    q:= nextprime(q)
  od:
  q
end proc:
seq(a(ithprime(n)), n=1..100); # Robert Israel, Nov 24 2014

Mathematica

Table[Function[p, q = 3; While[! Divisible[q^(p - 1) - 1, p^2], q = NextPrime@ q]; q]@ Prime@ n, {n, 54}] (* Michael De Vlieger, Feb 12 2017 *)

PARI

a(n) = {p = prime(n); forprime(q=3, , if (Mod(q, p^2)^(p-1) == 1, return (q)););} \\ Michel Marcus, Nov 24 2014

Extensions

Removed an incorrect comment. - Felix Fröhlich, Feb 12 2017