A218014 Location of the n-th prime in its Andrica ranking.

27, 6, 13, 1, 31, 4, 54, 8, 3, 100, 5, 25, 155, 28, 9, 16, 243, 19, 49, 288, 21, 62, 24, 12, 75, 422, 81, 444, 84, 2, 112, 37, 580, 11, 634, 47, 53, 150, 57, 60, 788, 20, 840, 183, 872, 10, 14, 218, 1029, 228, 80, 1074, 26, 87, 92, 99, 1237, 103, 281, 1319, 29, 15, 314, 1498, 323

Offset: 1

Author

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

Keywords

• nonn

Comments

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

Last appearance by prime index: 1, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, ..., .

Last appearance of a minimum prime by Andrica ranking: 2, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, ..., .

As expected, this sequence is the lesser of the twin primes beginning with the second term, 11. See A001359.

Examples

			a(1)=27 since the first prime, 2, does not show up in the ranking until the 27th term. See A218013.
a(4)=1 since the fourth prime, 7, has the maximum A_n value, see A218012; i.e., sqrt(p_n)-sqrt(p_n+1) is at a maximum.
		

Links

• Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT], 2010.

Crossrefs

Cf. A079296, A218012, A218015.

Programs

Mathematica

lst = {}; p = 2; q = 3; While[p < 1600000, If[ Sqrt[q] - Sqrt[p] > 1/20, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; lsu = First@ Transpose@ Sort[lst, #1[[2]] > #2[[2]] &]; Table[ Position[lsu, p, 1, 1], {p, Prime@ Range@ 65}] // Flatten