A218015 Number of primes p such that sqrt(q) - sqrt(p) > 1/n, where q is the prime after p.

0, 6, 22, 41, 75, 132, 186, 258, 330, 416, 511, 613, 724, 860, 1001, 1163, 1372, 1563, 1751, 1965, 2179, 2412, 2685, 2945, 3258, 3581, 3885, 4194, 4525, 4857, 5246, 5644, 6024, 6402, 6767, 7229, 7695, 8177, 8666, 9156, 9674, 10185, 10740, 11283, 11824

Offset: 1

Author

Marek Wolf and Robert G. Wilson v, Oct 18 2012

Keywords

• nonn

Comments

Also, the number of terms by Andrica ranking which are greater than 1/n.

Examples

			a(1) = 6 because only the primes 3, 7, 13, 23, 31 and 113 satisfy the criterion.
As an example, - sqrt(3) + sqrt(5) ~= 0.50401717 which is greater than 1/2.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..200
• Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT], 2010.

Crossrefs

Cf. A079296, A218012, A218014.

Programs

Mathematica

lst = {}; p = 2; q = 3; While[p < 10^8, If[ Sqrt[q] - Sqrt[p] > 1/50, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; Table[ Length@ Select[ lst, #[[2]] > 1/n &], {n, 50}]
nn = 50; t = Table[0, {nn}]; p = 2; q = 3; While[p < 10^8, n = Floor[1/(Sqrt[q] - Sqrt[p])]; If[n <= nn, t[[n]]++]; p = q; q = NextPrime[q]]; Join[{0}, Accumulate[t]] (* T. D. Noe, Oct 18 2012 *)