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 A237453 Number of primes p < n with p*n + pi(p) prime, where pi(.) is given by A000720. 6
 0, 0, 1, 0, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 3, 2, 1, 2, 1, 2, 3, 3, 2, 3, 1, 1, 1, 3, 2, 4, 3, 3, 3, 2, 1, 2, 1, 1, 3, 3, 1, 2, 3, 3, 3, 4, 3, 3, 2, 2, 6, 4, 3, 5, 3, 2, 3, 2, 4, 4, 3, 1, 3, 5, 2, 5, 3, 1, 2, 3, 2, 4, 2, 3, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 for no n > 144. Moreover, for any positive integer n, there is a prime p < sqrt(2*n)*log(5n) with p*n + pi(p) prime. (ii) For each integer n > 8, there is a prime p <= n + 1 with (p-1)*n - pi(p-1) prime. (iii) For every n = 1, 2, 3, ... there is a positive integer k < 3*sqrt(n) with k*n + prime(k) prime. (iv) For each n > 13, there is a positive integer k < n with k*n + prime(n-k) prime. We have verified that a(n) > 0 for all n = 5, ..., 10^8. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014 EXAMPLE a(3) = 1 since 2 and 2*3 + pi(2) = 6 + 1 = 7 are both prime. a(10) = 1 since 5 and 5*10 + pi(5) = 50 + 3 = 53 are both prime. a(107) = 1 since 89 and 89*107 + pi(89) = 9523 + 24 = 9547 are both prime. a(144) = 1 since 59 and 59*144 + pi(59) = 8496 + 17 = 8513 are both prime. MATHEMATICA a[n_]:=Sum[If[PrimeQ[Prime[k]*n+k], 1, 0], {k, 1, PrimePi[n-1]}] Table[a[n], {n, 1, 80}] PROG (PARI) vector(100, n, sum(k=1, primepi(n-1), isprime(prime(k)*n+k))) \\ Colin Barker, Feb 08 2014 CROSSREFS Cf. A000040, A000720, A233296, A237284, A237291, A237496, A237497. Sequence in context: A144790 A090996 A309736 * A265754 A089309 A126387 Adjacent sequences:  A237450 A237451 A237452 * A237454 A237455 A237456 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 08 2014 STATUS approved

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Last modified July 12 06:48 EDT 2020. Contains 335657 sequences. (Running on oeis4.)