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 A261353 Least positive integer k such that prime(prime(k))*prime(prime(k*n)) = prime(p)+2 for some prime p. 4
 11, 2, 1, 606, 350, 166, 53, 1865, 7, 45, 1308, 68, 215, 61, 256, 13, 248, 90, 1, 1779, 796, 1, 4, 444, 650, 55, 157, 303, 82, 84, 25, 3, 1912, 621, 128, 205, 164, 1091, 61, 12, 337, 1, 303, 15, 23, 418, 212, 23, 2494, 1, 472, 771, 1, 36, 8, 46, 8, 18, 264, 22, 725, 85, 65, 231, 606, 3, 1, 43, 144, 164 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(prime(m))*prime(prime(n)) = prime(p)+2 for some prime p. This implies that the sequence A261352 has infinitely many terms. REFERENCES 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. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..382 Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014. EXAMPLE a(1) = 11 since prime(prime(11))*prime(prime(11*1)) = prime(31)^2 = 127^2 = 16129 = prime(1877)+2 with 1877 prime. a(4) = 606 since prime(prime(606))*prime(prime(606*4)) = prime(4457)*prime(21589) = 42643*244471 = 10424976853 = prime(473490161)+2 with 473490161 prime. MATHEMATICA f[n_]:=Prime[Prime[n]] PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]] Do[k=0; Label[bb]; k=k+1; If[PQ[f[k]*f[k*n]-2], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}] CROSSREFS Cf. A000040, A109611, A261282, A261352. Sequence in context: A193925 A010190 A323454 * A087774 A322562 A040118 Adjacent sequences:  A261350 A261351 A261352 * A261354 A261355 A261356 KEYWORD nonn AUTHOR Zhi-Wei Sun, Aug 15 2015 STATUS approved

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Last modified December 12 22:06 EST 2019. Contains 329963 sequences. (Running on oeis4.)