A238278 a(n) = |{0 < k < n: the number of primes in the interval ((k-1)*n, k*n] and the number of primes in the interval (k*n, (k+1)*n] are both prime}|.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 22 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 3.

(ii) For any integer n > 3, there is a prime p < n such that the number of primes in the interval ((p-1)*n, p*n) is a prime.

We have verified part (i) for n up to 150000.

See also A238277 and A238281 for related conjectures.

Examples

			a(17) = 1 since the interval (9*17, 10*17] contains exactly 3 primes with 3 prime, and the interval (10*17, 11*17] contains exactly 3 primes with 3 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..3500
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A237578, A237643, A237705, A238224, A238277, A238281.

Programs

Mathematica

d[k_,n_]:=PrimePi[k*n]-PrimePi[(k-1)n]
a[n_]:=Sum[If[PrimeQ[d[k,n]]&&PrimeQ[d[k+1,n]],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]