cp's OEIS Frontend

A237453 Number of primes p < n with p*n + pi(p) prime, where pi(.) is given by A000720.

%I A237453 #33 Apr 06 2014 22:16:29
%S A237453 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,
%T A237453 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,
%U A237453 5,2,5,3,1,2,3,2,4,2,3,2
%N A237453 Number of primes p < n with p*n + pi(p) prime, where pi(.) is given by A000720.
%C A237453 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.
%C A237453 (ii) For each integer n > 8, there is a prime p <= n + 1 with (p-1)*n - pi(p-1) prime.
%C A237453 (iii) For every n = 1, 2, 3, ... there is a positive integer k < 3*sqrt(n) with k*n + prime(k) prime.
%C A237453 (iv) For each n > 13, there is a positive integer k < n with k*n + prime(n-k) prime.
%C A237453 We have verified that a(n) > 0 for all n = 5, ..., 10^8.
%H A237453 Zhi-Wei Sun, <a href="/A237453/b237453.txt">Table of n, a(n) for n = 1..10000</a>
%H A237453 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A237453 a(3) = 1 since 2 and 2*3 + pi(2) = 6 + 1 = 7 are both prime.
%e A237453 a(10) = 1 since 5 and 5*10 + pi(5) = 50 + 3 = 53 are both prime.
%e A237453 a(107) = 1 since 89 and 89*107 + pi(89) = 9523 + 24 = 9547 are both prime.
%e A237453 a(144) = 1 since 59 and 59*144 + pi(59) = 8496 + 17 = 8513 are both prime.
%t A237453 a[n_]:=Sum[If[PrimeQ[Prime[k]*n+k],1,0],{k,1,PrimePi[n-1]}]
%t A237453 Table[a[n],{n,1,80}]
%o A237453 (PARI) vector(100, n, sum(k=1, primepi(n-1), isprime(prime(k)*n+k))) \\ _Colin Barker_, Feb 08 2014
%Y A237453 Cf. A000040, A000720, A233296, A237284, A237291, A237496, A237497.
%K A237453 nonn
%O A237453 1,5
%A A237453 _Zhi-Wei Sun_, Feb 08 2014