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

1,1

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.

References

Dickson, History of the theory of numbers

Links

Table of n, a(n) for n=1..50.

Example

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.

Crossrefs

Cf. A145354.

Sequence in context: A045317 A215378 A078763 * A234530 A235632 A085306

Adjacent sequences: A157881 A157882 A157883 * A157885 A157886 A157887

Keyword

nonn

Author

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

Extensions

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

Status

approved