A293833 Number of primes p with A020330(n) < p < A020330(n+1).

2, 2, 5, 3, 2, 2, 14, 4, 3, 3, 4, 1, 4, 3, 45, 3, 6, 6, 6, 5, 3, 6, 4, 5, 5, 6, 3, 5, 4, 6, 140, 12, 5, 9, 8, 11, 8, 5, 8, 8, 12, 8, 9, 7, 7, 8, 7, 6, 7, 9, 10, 5, 8, 11, 9, 8, 8, 7, 7, 9, 9, 7, 471, 14, 12, 15, 17, 15, 14, 13, 15, 14, 17, 12, 16, 16, 9, 17, 14, 12

Offset: 1

Author

Zhi-Wei Sun, Oct 16 2017

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 12.

The terms of A020330 are usually called "binary squares". Our conjecture is an analog of Legendre's conjecture that for each n = 1,2,3,... there is a prime between n^2 and (n+1)^2.

Those a(2^n-1) = pi(2*4^n+2^n) - pi(4^n) are relatively large, where pi(x) is the prime-counting function given by A000720.

We have verified that a(n) > 0 for all n = 1..2*10^7.

Examples

			a(1) = 2 since 5 and 7 are the only primes in the interval (A020330(1), A020330(2)) = (3, 10).
a(12) = 1 since 211 is the only prime greater than A020330(12) = 204 and smaller than A020330(13) = 221.
a(8191) = a(2^13 - 1) = pi(2^27 + 2^13) - pi(2^26) = 3646196.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Wikipedia, Legendre's conjecture

Crossrefs

Cf. A000040, A000720, A014085, A020330.

Programs

Mathematica

f[n_]:=f[n]=(2^(Floor[Log[2,n]]+1)+1)*n;
a[n_]:=a[n]=PrimePi[f[n+1]-1]-PrimePi[f[n]];
Table[a[n],{n,1,80}]