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A153477
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Primes p such that 2p+1 and 2p^2+4p+1 are also prime.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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Select[Prime[Range[1500]], And@@PrimeQ[{2#+1, 2#^2+4#+1}]&] (* Harvey P. Dale, Sep 23 2012 *)
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PROG
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(Magma) [p: p in PrimesUpTo(12200) | IsPrime(2*p+1) and IsPrime(2*p^2+4*p+1) ];
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CROSSREFS
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Cf. A005384 (Sophie Germain primes p: 2p+1 is also prime).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Edited, corrected (2 added) and extended beyond a(8) by Klaus Brockhaus, Jan 01 2009
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STATUS
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approved
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