A237712 a(n) = |{0 < k < n: k*n + pi(k*n) is prime}|, where pi(.) is given by A000720.

0, 1, 1, 1, 0, 1, 3, 1, 3, 1, 2, 3, 4, 3, 3, 2, 2, 4, 4, 1, 5, 2, 2, 4, 2, 6, 8, 5, 6, 3, 4, 5, 2, 4, 3, 3, 8, 5, 8, 6, 4, 3, 10, 6, 6, 5, 1, 7, 4, 4, 6, 9, 6, 9, 5, 4, 6, 10, 3, 7, 7, 6, 3, 8, 13, 5, 8, 3, 9, 11, 4, 8, 6, 8, 11, 11, 11, 12, 13, 12, 10, 6, 7, 7, 4, 16, 10, 8, 9, 4, 6, 14, 11, 7, 4, 13, 10, 13, 8, 10

Offset: 1

Author

Zhi-Wei Sun, Feb 24 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 5.

This implies that there are infinitely many positive integers m with m + pi(m) prime.

Examples

			a(6) = 1 since 2*6 + pi(2*6) = 12 + 5 = 17 is prime.
a(47) = 1 since 21*47 + pi(21*47) = 987 + 166 = 1153 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2500
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000720, A237578.

Programs

Mathematica

p[n_]:=PrimeQ[n+PrimePi[n]]
a[n_]:=Sum[If[p[k*n],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]