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

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

Offset: 1

Author

Zhi-Wei Sun, Feb 08 2014

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A000040, A000720, A233296, A237284, A237291, A237496, A237497.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[Prime[k]*n+k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

PARI

vector(100, n, sum(k=1, primepi(n-1), isprime(prime(k)*n+k))) \\ Colin Barker, Feb 08 2014