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A153477
Primes p such that 2p+1 and 2p^2+4p+1 are also prime.
1
2, 3, 5, 23, 41, 131, 191, 293, 443, 653, 719, 1031, 1409, 1451, 1973, 2063, 2273, 2753, 3023, 3593, 3911, 4349, 4391, 4793, 5003, 5039, 5081, 5171, 5231, 5333, 5501, 6053, 6113, 7433, 7541, 7643, 8273, 8741, 8969, 9371, 10691, 10709, 11321, 11909, 12119
OFFSET
1,1
COMMENTS
Subsequence of A005384.
If p = 3*2(m-1)-1, q = 2*p+1 and r=2*p^2+4*p+1 (m>1), then p*q*2^m and r*2^m are amicable numbers (A063990), this follows immediately from Thabit ibn Kurrah theorem. - Vincenzo Librandi, Sep 30 2013
LINKS
Eric Weisstein's World of Mathematics, Thâbit ibn Kurrah Rule.
EXAMPLE
For prime p = 5, 2p+1 = 11 is prime and 2p^2+4p+1 = 71 is prime; for p=293, 2p+1 = 587 is prime and 2p^2+4p+1 = 172871 is prime.
For p=5=3*2-1, q=11, r=71, we have 5*11*4=220 and 71*4=284, which are amicable numbers. - Vincenzo Librandi, Sep 30 2013
MAPLE
a := proc (n) if isprime(n) = true and isprime(2*n+1) = true and isprime(2*n^2+4*n+1) = true then n else end if end proc: seq(a(n), n = 1 .. 13000); # Emeric Deutsch, Jan 02 2009
MATHEMATICA
Select[Prime[Range[1500]], And@@PrimeQ[{2#+1, 2#^2+4#+1}]&] (* Harvey P. Dale, Sep 23 2012 *)
PROG
(Magma) [p: p in PrimesUpTo(12200) | IsPrime(2*p+1) and IsPrime(2*p^2+4*p+1) ];
CROSSREFS
Cf. A005384 (Sophie Germain primes p: 2p+1 is also prime).
Sequence in context: A215317 A104736 A090710 * A080016 A171432 A214703
KEYWORD
nonn
AUTHOR
Vincenzo Librandi, Dec 27 2008
EXTENSIONS
Edited, corrected (2 added) and extended beyond a(8) by Klaus Brockhaus, Jan 01 2009
Extended by Emeric Deutsch, Jan 02 2009
STATUS
approved