A218014 -id:A218014 - cp's OEIS frontend

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

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

nonn

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

			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.

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.

Cf. A079296, A218012, A218014.

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 *)

A218012 Decimal expansion of -sqrt(7) + sqrt(11), Andrica's Maximum A_n.

Original entry on oeis.org

6, 7, 0, 8, 7, 3, 4, 7, 9, 2, 9, 0, 8, 0, 9, 2, 5, 8, 6, 1, 3, 3, 1, 6, 9, 8, 3, 0, 3, 1, 4, 2, 6, 2, 5, 8, 2, 1, 6, 8, 2, 9, 3, 6, 2, 5, 0, 6, 9, 0, 3, 4, 1, 6, 6, 9, 0, 3, 4, 7, 6, 8, 6, 9, 1, 5, 4, 1, 5, 8, 1, 9, 3, 7, 8, 7, 6, 0, 2, 1, 8, 9, 4, 8, 4, 5, 0, 5, 1, 2, 6, 5, 3, 7, 4, 7, 0, 4, 0, 2, 9, 1, 9, 4, 7

Offset: 0

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

For each consecutive prime pair, p and q with p < q, d = -sqrt(p) + sqrt(q) is unique. Place d in order from greatest to least and specify p. This is the maximum d.

			0.670873479290809258613316983031426258216829362506903416690347686915...

Titu Andreescu and Dorin Andrica, Number Theory, Structures, Examples, and Problems.
Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT] (2010).

Cf. A079296, A218014, A218015.

RealDigits[- Sqrt[7] + Sqrt[11], 10, 111][[1]]

cp's OEIS Frontend

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

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