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 A237705 Number of primes p < n with pi(n-p) prime, where pi(.) is given by A000720. 14
 0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 2, 1, 2, 3, 2, 3, 3, 2, 3, 3, 2, 4, 4, 3, 3, 1, 1, 3, 3, 2, 2, 1, 2, 6, 6, 5, 5, 4, 3, 5, 5, 4, 5, 5, 4, 6, 6, 6, 6, 3, 3, 5, 5, 5, 5, 2, 2, 5, 5, 3, 4, 5, 4, 8, 8, 3, 3, 1, 2, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 12, 26, 27, 32, 68. (ii) For any integer n > 5, there is a prime p <= n with pi(n+p) prime. (iii) If n > 32, then pi((n-p)^2) is prime for some prime p < n. Also, for each n > 6 there is an odd prime p < 2*n with pi((n - (p-1)/2)^2) prime. (iv) Any integer n > 11 can be written as p + q with p and pi(q^2 + q + 1) both prime. (v) Each integer n > 34 can be written as k + m with k and m positive integers such that pi(k^2) and pi(2*m^2) are both prime. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(5) = 1 since 2 and pi(5-2) = pi(3) = 2 are both prime. a(12) = 1 since 7 and pi(12-7) = pi(5) = 3 are both prime. a(15) = 2 since 3 and pi(15-3) = pi(12) = 5 are both prime, and 11 and pi(15-11) = pi(4) = 2 are both prime. a(26) = 1 since 23 and pi(26-23) = 2 are both prime. a(27) = 1 since 23 and pi(27-23) = 2 are both prime. a(32) = 1 since 29 and pi(32-29) = 2 are both prime. a(68) = 1 since 37 and pi(68-37) = pi(31) = 11 are both prime. MATHEMATICA q[n_]:=PrimeQ[PrimePi[n]] a[n_]:=Sum[If[q[n-Prime[k]], 1, 0], {k, 1, PrimePi[n-1]}] Table[a[n], {n, 1, 70}] CROSSREFS Cf. A000040, A000720, A237284, A237578, A237582. Sequence in context: A254296 A248371 A237768 * A340673 A031284 A064569 Adjacent sequences:  A237702 A237703 A237704 * A237706 A237707 A237708 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 11 2014 STATUS approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)