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A258803
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Least prime p such that n = (prime(q)-1)/(prime(p)-1) for some prime q.
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10
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2, 2, 5, 3, 2, 13, 11, 2, 23, 3, 11, 29, 19, 397, 2, 67, 131, 31, 5, 2, 5, 7, 1039, 5, 7, 67, 3, 787, 2, 13, 83, 149, 2, 89, 47, 43, 31, 809, 3, 5, 2, 307, 5, 61, 41, 5, 67, 19, 11, 1447, 101, 13, 881, 2, 37, 31, 331, 11, 1033, 3, 19, 839, 2, 61, 163, 59, 41, 1163, 3, 353, 67, 7, 313, 11, 7, 7, 101, 2, 71, 19, 7, 127, 409, 53, 149, 401, 283, 3, 2, 191, 43, 157, 163, 13, 2, 31, 89, 19, 5, 3
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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. Moreover, for any integers s and t with |s| = |t| = 1, each positive rational number r can be written as (prime(p) + s)/(prime(q) + t) with p and q both prime. - Sun
I have verified the conjecture for all those rational numbers r = a/b with a, b = 1, ..., 500. - Zhi-Wei Sun, Jun 13 2015
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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 1 = (prime(2) - 1)/(prime(2) - 1) with 2 prime.
a(2) = 2 since 2 = (prime(3) - 1)/(prime(2) - 1) with 2 and 3 both prime.
a(14) = 397 since 14 = (prime(4021) - 1)/(prime(397) - 1) = (38053 - 1)/(2719 - 1) with 379 and 4021 both prime.
a(23) = 1039 since 23 = (prime(17209) - 1)/(prime(1039) - 1) = (190579 - 1)/(8287 - 1) with 1039 and 17209 both prime.
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MATHEMATICA
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PQ[n_]:=PrimeQ[n] && PrimeQ[PrimePi[n]];
Do[k = 0; Label[bb]; k = k + 1; If[PQ[n * (Prime[Prime[k]] - 1) + 1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 100}] (* Sun *)
pq[n_] := PrimeQ@n && PrimeQ@ PrimePi@ n; a[n_] := Block[{k = 1}, While[!pq[1 + n*(Prime@ Prime@ k - 1)], k++]; Prime@k]; Array[a, 100] (* Giovanni Resta, Jun 11 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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