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A053182
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Primes p such that p^2 + p + 1 is prime.
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18
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2, 3, 5, 17, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Roger Horn computed the first 776 terms of this sequence around 1961 to test (with Paul Bateman) their conjecture on the density of simultaneous primes in polynomials. [Charles R Greathouse IV, Apr 05 2011]
Starting with a(3)=5 all terms are of the form 6k-1, k in A147683. [Zak Seidov, Nov 10 2008]
Also: Primes p such that the sum of divisors of p^2 (sigma(p^2) = A000203(p^2) = p^2+p+1) is prime. [Claudio Meller, Apr 07 2011]
The generated prime numbers p^2 + p +1 are exactly A053183. [Bernard Schott, Dec 20 2012]
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LINKS
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M. F. Hasler, Table of n, a(n) for n = 1..2650 (Terms up to 500000)
Paul T. Bateman, Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367.
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MATHEMATICA
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f[n_]:=1+n+n^2; lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst, p]], {n, 6!}]; lst (* Vladimir Orlovsky, Jul 24 2009 *)
Select[Prime[Range[427]], PrimeQ[#^2+#+1]&] (* Bruno Berselli, Nov 08 2011 *)
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PROG
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(PARI) isA053182(n)=isprime(n) && isprime(n^2+n+1) \\ - Michael B. Porter, Apr 23 2010
(MAGMA) [p: p in PrimesUpTo(10000) | IsPrime(p^2+p+1)]; // Vincenzo Librandi, Aug 06 2010
(PARI) c=0; forprime(p=1, default(primelimit), isprime(p^2+p+1) & write("/tmp/b053182.txt", c++, " "p)) \\ - M. F. Hasler, Apr 07 2011
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CROSSREFS
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Cf. A053184, A065508, A091567, A147683, A188596.
Sequence in context: A118958 A189536 A163588 * A211972 A076706 A019350
Adjacent sequences: A053179 A053180 A053181 * A053183 A053184 A053185
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KEYWORD
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easy,nonn,changed
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AUTHOR
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Enoch Haga, Mar 01 2000
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EXTENSIONS
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List changed to cross-reference by Franklin T. Adams-Watters, May 12 2010
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STATUS
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approved
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