A290947 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.

7, 13, 37, 43, 61, 193, 211, 271, 307, 331, 601, 673, 727, 757, 823, 1063, 1297, 1447, 1597, 1621, 1657, 1693, 2113, 2281, 2347, 2437, 2503, 3001, 3067, 3271, 3733, 4093, 4201, 4957, 5581, 6073, 6607, 7321, 7333, 7723, 7867, 8287, 8581, 8647, 9643, 10243

Offset: 1

Author

Amiram Eldar, Aug 14 2017

Keywords

• nonn

Comments

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, ...

The generated triangular 3-Carmichael numbers are: 8911, 115921, 8134561, 14913991, 60957361, 6200691841, 8863329511, 24151953871, 39799655911, 53799052231, 585796503601, ...

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A002997, A087788, A290945.

Programs

Mathematica

seq = {}; Do[p1 = 6 k + 1; p2 = 3 p1 - 2; p3 = (p1*p2 + 1)/2;
If[AllTrue[{p1, p2, p3}, PrimeQ], AppendTo[seq, p1]], {k, 1,
  2000}]; seq

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