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A260208 Least prime p such that 2p*n+1 = prime(q*n) for some prime q. 1
2, 3, 2, 107, 271, 3, 3, 523, 17, 191, 73, 2707, 587, 2017, 19, 233, 57193, 7583, 9791, 7, 2111, 1373, 43, 109, 1283, 463, 8179, 25583, 7489, 1733, 9011, 7753, 7853, 887, 10141, 71, 1373, 7927, 509, 1433, 4513, 2399, 4211, 26407, 307, 2843, 58579, 3121, 5519, 38371 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: a(n) exists for any n > 0. In general, if a > 0 is even and b is 1 or -1, then for any positive integer n there are primes p and q such that a*p*n+b = prime(q*n).

REFERENCES

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.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

EXAMPLE

a(5) = 271 since 2*271*5+1 = 2711 = prime(79*5) with 271 and 79 both prime.

MATHEMATICA

PQ[n_, p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]

Do[k=0; Label[aa]; k=k+1; If[PQ[n, 2*Prime[k]*n+1]], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]

CROSSREFS

Cf. A000040, A260197.

Sequence in context: A075121 A075108 A265590 * A242457 A190146 A026932

Adjacent sequences:  A260205 A260206 A260207 * A260209 A260210 A260211

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 19 2015

STATUS

approved

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Last modified September 21 04:58 EDT 2020. Contains 337267 sequences. (Running on oeis4.)