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 A238597 Number of primes p < 2*n with 2*pi(p) + 1 and p*(2n-1) - 2 both prime, where pi(.) is given by A000720. 2
 0, 1, 2, 1, 1, 2, 2, 4, 1, 1, 5, 3, 1, 3, 1, 2, 4, 3, 3, 2, 2, 3, 4, 3, 1, 5, 3, 1, 3, 2, 4, 5, 2, 2, 2, 3, 3, 6, 3, 3, 4, 2, 4, 5, 3, 4, 5, 3, 2, 6, 2, 3, 8, 1, 1, 5, 5, 3, 5, 4, 4, 6, 2, 3, 3, 4, 3, 7, 3, 1, 7, 4, 4, 5, 4, 3, 8, 4, 1, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 for no n > 195. (ii) For any integer n > 1, there is a prime p < 2*n with 2*pi(p) + 1 (or 2*pi(p) - 1) and 2*n + p both prime. Part (i) of this conjecture is an extension of the conjecture in A238580. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014. EXAMPLE a(9) = 1 since 5, 2*pi(5) + 1 = 2*3 + 1 = 7 and 5*(2*9-1) - 2 = 5*17 - 2 = 83 are all prime. a(28) = 1 since 3, 2*pi(3) + 1 = 2*2 + 1 = 5 and 3*(2*28-1) - 2 = 3*55 - 2 = 163 are all prime. a(195) = 1 since 71, 2*pi(71) + 1 = 2*20 + 1 = 41 and 71*(2*195-1) - 2 = 27617 are all prime. MATHEMATICA p[n_, k_]:=p[n, k]=PrimeQ[k]&&PrimeQ[2*PrimePi[k]+1]&&PrimeQ[k*(2n-1)-2] a[n_]:=a[n]=Sum[If[p[n, k], 1, 0], {k, 1, 2n-1}] Table[a[n], {n, 1, 80}] CROSSREFS Cf. A000040, A000720, A237284, A238580. Sequence in context: A116560 A103784 A153916 * A045870 A229037 A036863 Adjacent sequences:  A238594 A238595 A238596 * A238598 A238599 A238600 KEYWORD nonn AUTHOR Zhi-Wei Sun, Mar 01 2014 STATUS approved

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Last modified October 15 05:56 EDT 2018. Contains 316202 sequences. (Running on oeis4.)