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A259488
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Positive integers k with prime(k)+2 and prime(prime(k))+2 both prime.
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3
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2, 3, 7, 13, 296, 343, 395, 405, 408, 463, 469, 473, 542, 572, 577, 584, 625, 671, 673, 695, 837, 984, 1016, 1030, 1074, 1165, 1224, 1230, 1328, 1410, 1445, 1679, 1825, 1860, 1867, 1949, 2078, 2091, 2095, 2123, 2167, 2476, 2478, 2616, 2753, 2764, 2956, 3011, 3065, 3416, 3621, 3646, 3712, 3720, 3758, 3872, 3926, 4063, 4071, 4079, 4133, 4217, 4312, 4351, 4524, 4745, 4855, 4865, 4882, 4922
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OFFSET
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1,1
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COMMENTS
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The conjecture in A259487 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the current sequence has infinitely many terms.
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REFERENCES
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Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
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LINKS
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EXAMPLE
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a(1) = 2 since prime(2)+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(2) = 3 since prime(3)+2 = 7 and prime(prime(3))+2 = prime(7)+2 = 19 are both prime.
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MATHEMATICA
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n=0; Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2], n=n+1; Print[n, " ", k]], {k, 1, 5000}]
Select[Range[5000], AllTrue[{Prime[#]+2, Prime[Prime[#]]+2}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 18 2018 *)
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PROG
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(PARI) k=pk=0; forprime(ppk=2, 1e6, if(isprime(pk++), k++; if(isprime(pk+2) && isprime(ppk+2), print1(k", ")))) \\ Charles R Greathouse IV, Jun 29 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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