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.

3, 5, 11, 29, 53, 83, 173, 239, 281, 359, 431, 719, 761, 809, 911, 1031, 1103, 1223, 1289, 1451, 1481, 1511, 1559, 1931, 2069, 2339, 2351, 2393, 2693, 2699, 2819, 2969, 3359, 3491, 3539, 3851, 4019, 4211, 4409, 5039, 6113, 6269, 6329, 6491, 6521, 6551, 6581

Offset

1,1

Comments

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, ...

The generated Lucas-Carmichael numbers are 399, 2915, 63503, 2924099, 32148899, 192099599, 3603600899, 13105670399, 25027872803, ...

Subsequence of A005384 (Sophie Germain primes).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

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.

Mathematica

aQ[n_] := AllTrue[{n, 2n+1, 2 n^2+n-2}, PrimeQ]; Select[Range[10^3], aQ]
Select[Prime[Range[1000]], AllTrue[{2#+1, #(2#+1)-2}, PrimeQ]&] (* Harvey P. Dale, Aug 16 2024 *)

Prog

(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

Crossrefs

Cf. A006972, A216925, A290947, A292538.

Sequence in context: A045536 A319393 A019338 * A046134 A177932 A328329

Adjacent sequences: A292536 A292537 A292538 * A292540 A292541 A292542

Keyword

nonn

Author

Amiram Eldar, Sep 18 2017

Status

approved