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A053685 Primes p > 7 which are congruent to 2 or 4 (mod 5) for which 2p-1 is also prime.

%I A053685 #22 Jul 02 2025 16:01:59
%S A053685 19,37,79,97,139,157,199,229,307,337,367,379,439,499,547,577,607,619,
%T A053685 727,829,877,937,967,997,1009,1069,1237,1279,1297,1399,1429,1459,1609,
%U A053685 1627,1657,1759,1867,2029,2089,2137,2179,2467,2539,2557,2617,2707,2719
%N A053685 Primes p > 7 which are congruent to 2 or 4 (mod 5) for which 2p-1 is also prime.
%C A053685 For such primes p, 2p-1 divides Fibonacci(p). Actually it is also true that (2m-1) divides Fibonacci(m) for *all* m > 7, m = 2 or 4 (mod 5) for which 2m-1 is prime.
%C A053685 Intersection of A047211 and A005382 without terms <= 7. - _Reinhard Zumkeller_, Oct 03 2012
%H A053685 T. D. Noe, <a href="/A053685/b053685.txt">Table of n, a(n) for n = 1..1000</a>
%H A053685 Vladimir Drobot, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-1/drobot.pdf">On primes in the Fibonacci sequence</a>, Fib. Quart. 38 (1) (2000) 71.
%e A053685 Note that 19 is prime and so is 2*19-1 or 37.
%t A053685 okQ[n_]:=Module[{x=Mod[n,5]},PrimeQ[2n-1]&&MemberQ[{2,4},x]]; Select[Prime[Range[5,500]],okQ]  (* _Harvey P. Dale_, Jan 14 2011 *)
%o A053685 (Haskell)
%o A053685 a053685 n = a053685_list !! (n-1)
%o A053685 a053685_list = dropWhile (<= 7) $ i a047211_list a005382_list where
%o A053685    i xs'@(x:xs) ys'@(y:ys) | x < y     = i xs ys'
%o A053685                            | x > y     = i xs' ys
%o A053685                            | otherwise = x : i xs ys
%o A053685 -- _Reinhard Zumkeller_, Oct 03 2012
%Y A053685 Cf. A000045.
%K A053685 easy,nice,nonn
%O A053685 1,1
%A A053685 _James Sellers_, Feb 15 2000