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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 (list; graph; refs; listen; history; text; internal format)
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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

Adjacent sequences: A153474 A153475 A153476 * A153478 A153479 A153480

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

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Last modified November 27 22:56 EST 2022. Contains 358406 sequences. (Running on oeis4.)