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A222184 Primes p such that q^(p-1) == 1 (mod p^2) for some prime q < p. 5

%I

%S 11,43,59,71,79,97,103,137,263,331,349,359,421,433,487,523,653,659,

%T 743,859,863,907,919,983,1069,1087,1091,1093,1163,1223,1229,1279,1381,

%U 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

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

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

%C The smallest corresponding primes q are A222185.

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

%H Giovanni Resta, <a href="/A222184/b222184.txt">Table of n, a(n) for n = 1..10000</a>

%H W. Keller and J. Richstein, <a href="http://web.archive.org/web/20091109011757/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html">Fermat quotients that are divisible by p</a>.

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

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

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

%o (PARI) 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

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

%K nonn

%O 1,1

%A _Jonathan Sondow_, Feb 11 2013

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Last modified January 22 07:39 EST 2020. Contains 331139 sequences. (Running on oeis4.)