A290487 Prime numbers p such that q = p^2+p-1 and r = (p^3+p^2-p+1)/2 are also primes, and thus p*q*r is an extreme 3-Carmichael number.

3, 5, 31, 41, 83, 131, 349, 373, 401, 449, 463, 1483, 1801, 1871, 2003, 2011, 2551, 2711, 2861, 3229, 3739, 3779, 4111, 4493, 5413, 5659, 6323, 6421, 7309, 7369, 7523, 8101, 8713, 9091, 9161, 9239, 9719, 9733, 10691, 10709, 11941, 12781, 12893, 15101, 15959

Offset

1,1

Comments

Carlos Rivera said that a 3-Carmichael number is "extreme" if its prime factors are the triple (p, p^2+p-1, (p^3+p^2-p+1)/2), since Richard Pinch and Chris Nash showed that given a prime p which is the least in the triple of the prime factors of a 3-Carmichael numbers, p*q*r, with p < q < r, then r <= (p^3+p^2-p+1)/2.

The corresponding extreme 3-Carmichael numbers are 561, 10585, 471905281, 2489462641, 167385219121, 2565387704881, 908658513945001, 1353761995707001, 2089268322242401, 4115059336080001, 4946817525914881, 5326010469307447921, 17081820793305370801, 21472202547687640321, 32321314831390144561,...

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1890 from Vincenzo Librandi)

Carlos Rivera, Conjecture 19, A bound to the largest prime factor of certain Carmichael numbers, The Prime Puzzles and Problems Connection.

Example

p = 5, q = p^2+p-1 = 29 and r = (p^3+p^2-p+1)/2 = 73 are all primes, thus 5 is in the sequence. 5*29*73 = 10585 is a 3-Carmichael number.

Mathematica

Select[Prime[Range[2 10^3]], And@@PrimeQ[{#^2 + # - 1, ((#^3 + #^2 - # + 1)/2)}]&] (* Vincenzo Librandi, Oct 13 2017 *)

Prog

(PARI) lista(nn) = forprime(p=3, nn, if (isprime(p^2+p-1) && isprime((p^3+p^2-p+1)/2), print1(p, ", "))); \\ Michel Marcus, Oct 13 2017
(Magma) [p: p in PrimesInInterval(3, 2*10^4) | IsPrime(p^2+p-1) and IsPrime((p^3+p^2-p+1) div 2)]; // Vincenzo Librandi, Oct 13 2017

Crossrefs

Cf. A087788 (3-Carmichael numbers).

Sequence in context: A189739 A372800 A035410 * A059940 A106893 A175097

Adjacent sequences: A290484 A290485 A290486 * A290488 A290489 A290490

Keyword

nonn

Author

Amiram Eldar, Aug 03 2017

Status

approved