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A257926
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Least positive integer k such that prime(k*n)+2 = prime(i*n)*prime(j*n) for some 0 < i < j.
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7
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6, 4, 10, 8, 451, 426, 622, 175, 1424, 500, 33, 703, 1761, 4428, 1563, 959, 8147, 7055, 5948, 250, 7517, 12706, 8405, 2948, 2610, 1949, 10424, 2214, 6722, 1963, 3335, 16382, 15687, 17591, 15073, 7818, 32202, 31169, 2248, 14899, 69955, 7580, 2393, 39295, 42352, 5884, 9367, 3630, 14090, 1305
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) exists for any n > 0.
This is much stronger than Chen's famous result that there are infinitely many Chen primes.
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REFERENCES
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Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
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) = 6 since prime(6*1)+2 = 15 = 3*5 = prime(2*1)*prime(3*1).
a(3) = 10 since prime(10*3)+2 = 115 = 5*23 = prime(1*3)*prime(3*3).
a(149) = 1476387 since prime(1476387*149)+2 = 4666119529 = 8311*561439 = prime(7*149)*prime(310*149).
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MATHEMATICA
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Dv[n_]:=Divisors[Prime[n]+2]
L[n_]:=Length[Dv[n]]
P[k_, n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n], 2]]&&Mod[PrimePi[Part[Dv[k*n], 2]], n]==0&&PrimeQ[Part[Dv[k*n], 3]]&&Mod[PrimePi[Part[Dv[k*n], 3]], n]==0
Do[k=0; Label[bb]; k=k+1; If[P[k, n], Goto[aa]]; Goto[bb]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]
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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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