A238281 a(n) = |{0 < k < n: the two intervals (k*n, (k+1)*n) and ((k+1)*n, (k+2)*n) contain the same number of primes}|.

0, 1, 2, 1, 2, 3, 3, 1, 5, 2, 4, 4, 8, 3, 7, 4, 4, 4, 2, 3, 7, 3, 10, 4, 12, 7, 7, 15, 7, 9, 8, 5, 8, 9, 11, 8, 8, 10, 8, 4, 10, 10, 10, 11, 7, 10, 8, 11, 8, 8, 9, 9, 8, 11, 7, 8, 13, 10, 8, 14, 13, 4, 14, 8, 11, 12, 14, 12, 8, 10, 16, 12, 16, 12, 14, 19, 11, 14, 8, 9

Offset: 1

Author

Zhi-Wei Sun, Feb 22 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, if n > 1 is not equal to 8, then there is a positive integer k < n with 2*k + 1 prime such that the two intervals ((k-1)*n, k*n) and (k*n, (k+1)*n) contain the same number of primes.

(ii) For any integer n > 4, there is a positive integer k < prime(n) such that all the three intervals (k*n, (k+1)*n), ((k+1)*n, (k+2)*n), ((k+2)*n, (k+3)*n) contain the same number of primes, i.e., pi(k*n), pi((k+1)*n), pi((k+2)*n), pi((k+3)*n) form a 4-term arithmetic progression.

Examples

			a(8) = 1 since each of the two intervals (7*8, 8*8) and (8*8, 9*8) contains exactly two primes.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000720, A237578, A238224, A238277, A238278.

Programs

Mathematica

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