A245659 Prime numbers P such that Q=2*P^2-1, R=2*Q^2-1, S=2*R^2-1 and T=2*S^2-1 are all prime numbers.

281683, 496789, 823421, 1352753, 1719217, 6174109, 8643149, 9761051, 9843529, 16191167, 19132121, 19745797, 23490473, 28457797, 31820429, 32860271, 36552277, 37068569, 43506569, 44776981, 46808903, 55035047, 55957807, 67194403, 75099137, 83092897, 86580421, 89135089

Offset

1,1

Comments

Subsequence of A106483.

For P = 496789, 83092897, 467014643, U=2*T^2-1 is also prime. [Corrected by Jens Kruse Andersen, Aug 21 2014]

Links

Pierre CAMI, Table of n, a(n) for n = 1..140

Example

281683 is prime P.
Q=2*P^2-1 = 158690624977 is prime Q.
R=2*Q^2-1 = 50365428911181712501057 is prime R.
S=2*R^2-1 = 5073352858814597404058971422301788780452234497 is prime S.
T=2*S^2-1 = 51477818460084496601334991724899650493354568309112026195311592373475872924903206720553686017 is prime T.
U=2*T^2-1 is composite.

Mathematica

f[n_]:=2n^2-1; Select[Prime[Range[5170000]], PrimeQ[f[#]]&&PrimeQ[f[f[#]]]&&PrimeQ[f[f[f[#]]]]&&PrimeQ[f[f[f[f[#]]]]]&] Farideh Firoozbakht, Aug 11 2014
Select[Prime[Range[52*10^5]], AllTrue[Rest[FoldList[2#^2-1&, {#, #, #, #, #}]], PrimeQ]&] (* Harvey P. Dale, Jan 13 2023 *)

Prog

(PFGW & SCRIPT)
SCRIPT
DIM i
DIM j
DIM k
DIM n, 0
DIMS t
OPENFILEOUT myf, a(n).txt
LABEL loop1
SET n, n+1
SETS t, %d, %d\,; n; p(n)
PRP 2*p(n)^2-1, t
IF ISPRP THEN GOTO a
GOTO loop1
LABEL a
SET i, 2*p(n)^2-1
PRP 2*i^2-1, t
IF ISPRP THEN GOTO b
GOTO loop1
LABEL b
SET j, 2*i^2-1
PRP 2*j^2-1, t
IF ISPRP THEN GOTO c
GOTO loop1
LABEL c
WRITE myf, t
SET k, 2*j^2-1
PRP 2*k^2-1
IF ISPRP THEN GOTO d
GOTO loop1
LABEL d
WRITE myf, t
GOTO loop1
(PARI)
f(x)=return(2*x^2-1)
forprime(p=1, 10^8, if(ispseudoprime(f(p)) && ispseudoprime(f(f(p))) && ispseudoprime(f(f(f(p)))) && ispseudoprime(f(f(f(f(p))))), print1(p, ", "))) \\ Derek Orr, Jul 28 2014

Crossrefs

Cf. A106483.

Sequence in context: A254740 A131263 A235173 * A321055 A185886 A235827

Adjacent sequences: A245656 A245657 A245658 * A245660 A245661 A245662

Keyword

nonn

Author

Pierre CAMI, Jul 28 2014

Extensions

More terms from Derek Orr, Jul 28 2014

Status

approved