# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a319040 Showing 1-1 of 1 %I A319040 #36 Sep 16 2018 04:42:04 %S A319040 7,17,23,31,35,41,47,71,73,79,89,97,103,113,127,137,151,167,169,191, %T A319040 193,199,223,233,239,241,257,263,271,281,311,313,337,353,359,367,383, %U A319040 385,401,409,431,433,439,449,457,463,479,487,503,521,569,577,593,599 %N A319040 Numbers k > 1 such that Pell(k) == 1 (mod k). %C A319040 It appears that most of the terms of this sequence are primes. The composite terms are 35, 169, 385, 899, 961, 1121, ... (A319042). %C A319040 The primes in the sequence give A001132 (primes == +-1 (mod 8)), since for primes p we have Pell(p) == (2/p) (mod p) where (2/p) is the Legendre symbol. - _Jianing Song_, Sep 10 2018 %e A319040 k = 7 is in the sequence since Pell(7) = 169 = 7 * 24 + 1 == 1 (mod 7). %e A319040 k = 11 is not in the sequence: Pell(11) = 5741 = 11 * 522 - 1 !== 1 (mod 11). %e A319040 k = 35 is in the sequence: Pell(35) = 8822750406821 = 35 * 252078583052 + 1 == 1 (mod 35). %p A319040 isA319040 := k -> simplify(2^(k-1)*hypergeom([1-k/2,(1-k)/2],[1-k],-1)) mod k = 1: A319040List := b -> select(isA319040, [$1..b]): %p A319040 A319040List(600); # _Peter Luschny_, Sep 09 2018 %t A319040 Select[Range[500], Mod[Fibonacci[#, 2], #] == 1 &] (* _Alonso del Arte_, Sep 08 2018 *) %Y A319040 Cf. A000129 (Pell numbers), A001132, A023173, A319041, A319042, A319043. %K A319040 nonn %O A319040 1,1 %A A319040 _Jon E. Schoenfield_, Sep 08 2018 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE