

A259487


Least positive integer m with prime(m)+2 and prime(prime(m))+2 both prime such that prime(m*n)+2 and prime(prime(m*n))+2 are both prime.


10



2, 1860, 408, 25011, 51312, 37977, 695, 4071, 10970, 3621, 17671, 12005, 1230, 19494, 542, 577, 408, 2476, 584, 542, 469, 34229, 343, 24078, 3011, 25749, 20706, 24198, 2478, 3926, 1030, 1030, 13857, 3621, 343, 13380, 2476, 4922, 2476, 296, 19176, 29175, 34737, 13, 625, 2956, 408, 572, 7, 469, 15604, 9699, 26515, 2167, 5302, 9773, 54254, 1410, 4524, 4351
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OFFSET

1,1


COMMENTS

Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259488.
This implies that there are infinitely many primes p with p+2 and prime(p)+2 both prime.
I have verified the conjecture for all those r = a/b with a,b = 1,...,400.  ZhiWei Sun, Jun 29 2015


REFERENCES

ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS



EXAMPLE

a(1) = 2 since prime(2)+2 = 3+2 = 5 and prime(prime(2))+2 = prime(3)+2 = 7 are both prime, but prime(1)+2 = 4 is composite.
a(49) = 7 since prime(7)+2 = 17+2 = 19, prime(prime(7))+2 = prime(17)+2 = 59+2 = 61, prime(49*7)+2 = 2309+2 = 2311 and prime(prime(49*7))+2 = prime(2309)+2 = 20441+2 = 20443 are all prime.


MATHEMATICA

PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]]+2]
Do[k=0; Label[bb]; k=k+1; If[PQ[k]&&PQ[n*k], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



