Numbers n such that the number of composite numbers around n^2 is a new record.

2, 3, 5, 11, 23, 30, 41, 50, 76, 100, 149, 159, 189, 345, 437, 509, 693, 1110, 1165, 5018, 14908, 18906, 19079, 28634, 38682, 80444, 105686, 185179, 265236, 269697, 409049

1

The number of composite numbers is the difference between the largest prime less than n^2 and the smallest prime greater than n^2. See S000385 for the record number of composite numbers.

T. D. Noe, Plot of 31 terms

(Mma) t = {}; mx = 0; Do[k = n^2; k2 = PrimePi[k]; diff = Prime[k2 + 1] - Prime[k2]; If[diff > mx, mx = diff; AppendTo[t, {n, diff}]], {n, 2, 10000}]; Transpose[t][[1]]

Cf. S000385.

nonn,hard

T. D. Noe, Nov 25 2014