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

1,2

Comments

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

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.

Example

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.

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 *)

Crossrefs

Cf. A079296, A218012, A218014.

Sequence in context: A264043 A056821 A031083 * A031305 A015721 A182200

Adjacent sequences: A218012 A218013 A218014 * A218016 A218017 A218018

Keyword

nonn

Author

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

Status

approved