OFFSET
1,2
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).
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.
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009
EXTENSIONS
More terms from Robert G. Wilson v, May 04 2009
STATUS
approved