This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A292539 #15 Aug 16 2024 18:10:05
%S A292539 3,5,11,29,53,83,173,239,281,359,431,719,761,809,911,1031,1103,1223,
%T A292539 1289,1451,1481,1511,1559,1931,2069,2339,2351,2393,2693,2699,2819,
%U A292539 2969,3359,3491,3539,3851,4019,4211,4409,5039,6113,6269,6329,6491,6521,6551,6581
%N A292539 Primes p1 such that p2 = 2p1 + 1 and p3 = p1*p2 - 2 are also primes, so p1*p2*p3 is a Lucas-Carmichael number of the form k^2 - 1.
%C A292539 All the primes, except the first, are of the form p1 = 6k - 1, p2 = 12k - 1, p3 = 72k^2 - 18k - 1, with k = 1, 2, 5, 9, 14, 29, 40, 47, 60, 72, 120, 127, 135, 152, 172, 184, ...
%C A292539 The generated Lucas-Carmichael numbers are 399, 2915, 63503, 2924099, 32148899, 192099599, 3603600899, 13105670399, 25027872803, ...
%C A292539 Subsequence of A005384 (Sophie Germain primes).
%H A292539 Amiram Eldar, <a href="/A292539/b292539.txt">Table of n, a(n) for n = 1..10000</a>
%e A292539 p1 = 3 is in the sequence since with p2 = 2*3 + 1 = 7 and p3 = 3*7 - 2 = 19 they are all primes. 3*7*19 = 399 is a Lucas-Carmichael number.
%t A292539 aQ[n_] := AllTrue[{n, 2n+1, 2 n^2+n-2}, PrimeQ]; Select[Range[10^3], aQ]
%t A292539 Select[Prime[Range[1000]],AllTrue[{2#+1,#(2#+1)-2},PrimeQ]&] (* _Harvey P. Dale_, Aug 16 2024 *)
%o A292539 (PARI) is(n) = if(!ispseudoprime(n), return(0), my(p=2*n+1); if(!ispseudoprime(p), return(0), if(ispseudoprime(n*p-2), return(1)))); 0 \\ _Felix Fröhlich_, Sep 18 2017
%Y A292539 Cf. A006972, A216925, A290947, A292538.
%K A292539 nonn
%O A292539 1,1
%A A292539 _Amiram Eldar_, Sep 18 2017