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A261362
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Least positive integer k such that both k and k*n belong to the set {m>0: 2*prime(prime(m))+1 = prime(p) for some prime p}.
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3
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2, 21531, 2, 35434, 11107, 35175, 24674, 64624, 127943, 1981, 155709, 50657, 74313, 11479, 6, 1981, 43405, 40859, 74229, 2, 154292, 51711, 29460, 29011, 42001, 28352, 2979, 85836, 6936, 186608, 3705, 14402, 25525, 96192, 6, 113433, 164, 787, 71873, 3365, 93169, 47219, 43128, 184740, 2, 78329, 13656, 6936, 139469, 26713
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OFFSET
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1,1
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COMMENTS
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Conjecture: Let a,b,c be positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. If a+b+c is even and a is not equal to b, then any positive rational number r can be written as m/n with m and n in the set {k>0: a*prime(p) - b*prime(prime(k)) = c for some prime p}.
This implies the conjecture in A261361.
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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(2) = 21531 since 2*prime(prime(21531))+1 = 2*prime(243799)+1 = 2*3403703+1 = 6807407 = prime(464351) with 464351 prime, and 2*prime(prime(21531*2))+1 = 2*prime(520019)+1 = 2*7686083+1 = 15372167 = prime(993197) with 993197 prime.
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MATHEMATICA
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f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0; Label[bb]; k=k+1; If[PQ[f[k]]&&PQ[f[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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