A222185 -id:A222185 - cp's OEIS frontend

A222184 Primes p such that q^(p-1) == 1 (mod p^2) for some prime q < p.

11, 43, 59, 71, 79, 97, 103, 137, 263, 331, 349, 359, 421, 433, 487, 523, 653, 659, 743, 859, 863, 907, 919, 983, 1069, 1087, 1091, 1093, 1163, 1223, 1229, 1279, 1381, 1483, 1499, 1549, 1657, 1663, 1667, 1697, 1747, 1777, 1787, 1789, 1877, 1993, 2011, 2213, 2221, 2251, 2281, 2309, 2371, 2393, 2473, 2671, 2719, 2777, 2791, 2803, 2833, 2861, 3037, 3079, 3163, 3251, 3257, 3463, 3511, 3557

Offset: 1

Jonathan Sondow, Feb 11 2013

nonn

Subsequence of A134307; see its interesting heuristics. (What is the analogous heuristic for the present sequence?)

The smallest corresponding primes q are A222185.

			3 is a prime < 11, and 11^2 divides 3^(11-1)-1 = 59048 = 121*488, so 11 is a member.

L. E. Dickson, History of the Theory of Numbers, vol. 1, chap. IV.

Giovanni Resta, Table of n, a(n) for n = 1..10000
W. Keller and J. Richstein, Fermat quotients that are divisible by p.

Cf. A001220, A039678, A134307, A143548, A222185.

Select[ Prime[ Range[500]], Product[ PowerMod[ Prime[k], # - 1, #^2] - 1, {k, 1, PrimePi[#] - 1}] == 0 &]

lista(nn) = {forprime (p=2, nn, ok = 0; forprime(q=2, p-1, if (Mod(q, p^2)^(p-1) == 1, ok=1; break);); if (ok, print1(p, ", ")););} \\ Michel Marcus, Nov 24 2014

A222185(n)^(a(n)-1) == 1 (mod a(n)^2).

A222206 Least prime p such that q^(p-1) == 1 (mod p^2) for n primes q < p.

Original entry on oeis.org

2, 11, 349, 13691, 24329

Offset: 0

Jonathan Sondow, Feb 12 2013

I found no new terms < 5*10^6. - J. Stauduhar, Mar 23 2013

a(5) > 13*10^6, if it exists. Note that, up to 13*10^6, the only other prime p (apart 24329) such that the congruence is satisfied for 4 primes q < p is 9656869. - Giovanni Resta, May 23 2017

			For the prime p = 349, but for no smaller prime, there are 2 primes q = 223 and 317 < p with  q^(p-1) == 1 (mod p^2), so a(2) = 349.

L. E. Dickson, History of the Theory of Numbers, vol. 1, chap. IV.

W. Keller and J. Richstein, Fermat quotients that are divisible by p. [Broken link]
Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod (p^r)), Math. Comp. 74 (2005), 927-936.

Cf. A001220, A039678, A134307, A143548, A222184, A222185.

f[n_] := Block[{p = 2, q = {}}, While[ Count[ PowerMod[ q, p - 1, p^2], 1] != n, q = Join[q, {p}]; p = NextPrime@ p]; p]; Array[f, 5, 0] (* Robert G. Wilson v, Mar 09 2015 *)

a(n) = {nb = 0; p = 2; while (nb != n, p = nextprime(p+1); nb = 0; forprime(q=2, p-1, if (Mod(q, p^2)^(p-1) == 1, nb ++); if (nb > n, break););); p;} \\ Michel Marcus, Mar 08 2015

cp's OEIS Frontend

A222184 Primes p such that q^(p-1) == 1 (mod p^2) for some prime q < p.

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