%I #10 Aug 14 2017 21:34:37
%S 7,13,37,43,61,193,211,271,307,331,601,673,727,757,823,1063,1297,1447,
%T 1597,1621,1657,1693,2113,2281,2347,2437,2503,3001,3067,3271,3733,
%U 4093,4201,4957,5581,6073,6607,7321,7333,7723,7867,8287,8581,8647,9643,10243
%N Primes p1 > 3, such that p2 = 3p1-2 and p3 = (p1*p2+1)/2 are also primes, so p1*p2*p3 is a triangular 3-Carmichael number.
%C The primes are of the form p1=(6k+1), p2=(18k+1), and p3=(54k^2+12k+1), with k = 1, 2, 6, 7, 10, 32, 35, 45, 51, 55, 100, ...
%C The generated triangular 3-Carmichael numbers are: 8911, 115921, 8134561, 14913991, 60957361, 6200691841, 8863329511, 24151953871, 39799655911, 53799052231, 585796503601, ...
%H Charles R Greathouse IV, <a href="/A290947/b290947.txt">Table of n, a(n) for n = 1..10000</a>
%e p1 = 7 is in the sequence since with p2 = 3*7-2 = 19 and p3 = (7*19+1)/2 = 67 they are all primes. 7*19*67 = 8911 is a triangular 3-Carmichael number.
%t seq = {}; Do[p1 = 6 k + 1; p2 = 3 p1 - 2; p3 = (p1*p2 + 1)/2;
%t If[AllTrue[{p1, p2, p3}, PrimeQ], AppendTo[seq, p1]], {k, 1,
%t 2000}]; seq
%o (PARI) list(lim)=my(v=List()); forprime(p=7,lim, if(isprime(3*p-2) && isprime((p*(3*p-2)+1)/2), listput(v,p))); Vec(v) \\ _Charles R Greathouse IV_, Aug 14 2017
%Y Cf. A000217, A002997, A087788, A290945.
%K nonn
%O 1,1
%A _Amiram Eldar_, Aug 14 2017
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