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

1,1

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, ...

Links

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

Example

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.

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

Prog

(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

Crossrefs

Cf. A000217, A002997, A087788, A290945.

Sequence in context: A117706 A066673 A307762 * A267985 A173230 A361483

Adjacent sequences: A290944 A290945 A290946 * A290948 A290949 A290950

Keyword

nonn

Author

Amiram Eldar, Aug 14 2017

Status

approved