OFFSET

1,3

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.

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

EXAMPLE

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

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}]

CROSSREFS

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 22 2014

STATUS

approved