A238277 a(n) = |{0 <= k < n: the number of primes in the interval (k*n, (k+1)*n] is a square}|.

1, 2, 2, 2, 2, 2, 2, 3, 1, 3, 2, 4, 1, 5, 3, 3, 10, 11, 8, 7, 10, 6, 13, 11, 13, 8, 12, 10, 8, 7, 7, 6, 4, 5, 5, 6, 3, 4, 7, 3, 7, 7, 8, 7, 7, 9, 8, 12, 8, 5, 12, 11, 14, 11, 14, 11, 8, 11, 9, 9, 13, 12, 5, 14, 15, 12, 15, 12, 15, 14, 15, 16, 13, 10, 18, 20, 12, 7, 17, 13

Offset: 1

Author

Zhi-Wei Sun, Feb 22 2014

Keywords

• nonn

Comments

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

We have verified this for n up to 10^5.

See also A238278 and A238281 for related conjectures.

Examples

			a(9) = 1 since the interval (0, 9] contains exactly 2^2 = 4 primes.
a(13) = 1 since the interval (9*13, 10*13] contains exactly 1^2 = 1 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000290, A237598, A237706, A238278, A238281.

Programs

Mathematica

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