A157884 For each positive integer m there exist at least one prime Q=Q(m) and at least one prime P=P(m) such that (2m-1)^2 < Q < (2m)^2 - (2m-1) <= P < (2m)^2. Sequence lists pairs P(m), Q(m) for m >= 1. If more than one prime for P or Q exists, we take the smallest.

2, 3, 11, 13, 29, 31, 53, 59, 83, 97, 127, 137, 173, 191, 227, 241, 293, 307, 367, 383, 443, 463, 541, 557, 631, 653, 733, 757, 853, 877, 967, 997, 1091, 1123, 1229, 1277, 1373, 1409, 1523, 1567, 1693, 1723, 1861, 1901, 2027, 2081, 2213, 2267, 2411, 2459

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 08 2009

Keywords

• nonn

Comments

In some intervals there is one prime only: Q(1)=2, P(1)=3, Q(2)=11, P(2)=13, Q(3)=29, P(3)=31, Q(4)=53, P(5)=97.

Second part of numerical results to the problem: There is always a prime p in the interval between two consecutive square numbers: n^2 <= p <= (n+1)^2.

Examples

			m=1: 1 < Q < 3 <= P < 4; the only such prime Q and the only such prime P are Q(1)=2 and P(1)=3, so a(1)=2, a(2)=3.
m=2: 9 < Q < 13 <= P < 16; the only such prime Q and the only such prime P are Q(2)=11 and P(2)=13, so a(3)=11, a(4)=13.
m=4: 49 < Q < 57 <= P < 64; the only such prime Q is Q(4)=53, but there are two such primes P (59 and 61), so we take the smaller one, thus P(4)=59, so a(7)=53, a(8)=59.
		

References

• Dickson, History of the theory of numbers

Crossrefs

Cf. A145354.

Extensions

277 replaced with 241, 347 with 307, 431 with 383, etc. by R. J. Mathar, Nov 01 2010