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A215850 Primes p such that 2*p + 1 divides Lucas(p).

%I A215850 #24 Feb 16 2025 08:33:18
%S A215850 5,29,89,179,239,359,419,509,659,719,809,1019,1049,1229,1289,1409,
%T A215850 1439,1499,1559,1889,2039,2069,2129,2339,2399,2459,2549,2699,2819,
%U A215850 2939,2969,3299,3329,3359,3389,3449,3539,3779,4019,4349,4409,4919,5039,5279,5399,5639
%N A215850 Primes p such that 2*p + 1 divides Lucas(p).
%C A215850 An equivalent definition of this sequence: 5 together with primes p such that p == -1 (mod 30) and 2*p + 1 is also prime.
%C A215850 Sequence without the initial 5 is the intersection of A005384 and A132236.
%C A215850 These numbers do not occur in A137715.
%C A215850 From _Arkadiusz Wesolowski_, Aug 25 2012: (Start)
%C A215850 The sequence contains numbers like 1409 which are in A053027.
%C A215850 a(n) is in A002515 if and only if a(n) is congruent to -1 mod 60. (End)
%H A215850 Arkadiusz Wesolowski, <a href="/A215850/b215850.txt">Table of n, a(n) for n = 1..10000</a>
%H A215850 C. K. Caldwell, "Top Twenty" page, <a href="https://t5k.org/top20/page.php?id=65">Lucas cofactor</a>
%H A215850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%e A215850 29 is in the sequence since it is prime and 59 is a factor of Lucas(29) = 1149851.
%t A215850 Select[Prime@Range[740], Divisible[LucasL[#], 2*# + 1] &]
%t A215850 Prepend[Select[Range[29, 5639, 30], PrimeQ[#] && PrimeQ[2*# + 1] &], 5]
%o A215850 (Magma) [5] cat [n: n in [29..5639 by 30] | IsPrime(n) and IsPrime(2*n+1)];
%o A215850 (PARI) is_A215850(n)=isprime(n)&!real((Mod(2,2*n+1)+quadgen(5))*quadgen(5)^n) \\ - _M. F. Hasler_, Aug 25 2012
%Y A215850 Supersequence of A230809. Cf. A000032, A132236.
%K A215850 nonn
%O A215850 1,1
%A A215850 _Arkadiusz Wesolowski_, Aug 24 2012