A159805 Number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.

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

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) Second ("right") half-interval: number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.

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

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

Except for a(1), this is a bisection of A094189 and hence related to a conjecture of Oppermann. - T. D. Noe, Apr 22 2009

Examples

			m=1: 3 <= p < 4 => prime 3: a(1)=1;
m=4: 57 <= p < 64 => primes 59,61: a(4)=2;
m=5: 91 <= p < 100 => prime 97: a(5)=1;
m=30: 3541 <= p < 3600 => primes 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593: 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

Maple

A159805 := proc(n) local a,p; a := 0 ; for p from 4*n^2-2*n+1 to 4*n^2-1 do if isprime(p) then a := a+1 ; fi; od: a ; end: seq(A159805(n),n=1..120) ; # R. J. Mathar, Apr 22 2009

Extensions

More terms from R. J. Mathar, Apr 22 2009