A159802 Number of primes q with (2m)^2+1 <= q < (2m+1)^2-2m.

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

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009

Keywords

• nonn

Comments

1) Immediate connection to unsolved problem, is there always a prime between n^2 and (n+1)^2 ("full" interval of two consecutive squares).

2) See sequence A145354 and A157884 for more details to this new improved conjecture.

3) First ("left") half interval, primes q with (2m)^2+1 <= q < (2m+1)^2-2m.

4) It is conjectured that a(m) >= 1.

5) No a(m) with m>1 is known, where a(m)=1.

This is a bisection of A089610 and hence related to a conjecture of Oppermann. - T. D. Noe, Apr 22 2009

Examples

			1) m=1: 5 <= q < 7 => prime 5: a(1)=1.
2) m=2: 17 <= q < 21 => primes 17, 19: a(2)=2.
3) m=3: 37 <= q < 43 => primes 37, 41: a(3)=2.
4) m=30: 3601 <= q < 3661 => primes 3607,3613,3617,3623,3631,3637,3643,3659: a(30)=8.
		

References

• L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
• R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
• P. Ribenboim, The New Book of Prime Number Records. Springer. 1996

Crossrefs

Cf. A145354, A157884, A014085.

Programs

Mathematica

f[n_] := PrimePi[(2 n + 1)^2 - 2 n - 1] - PrimePi[(2 n)^2]; Table[ f@n, {n, 85}] (* Robert G. Wilson v, May 04 2009 *)

Extensions

More terms from Robert G. Wilson v, May 04 2009