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

1,1

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.

Links

Table of n, a(n) for n=1..65.

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

Example

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.

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

Crossrefs

Cf. A079296, A218012, A218015.

Sequence in context: A175240 A204877 A040709 * A040710 A040708 A214103

Adjacent sequences: A218011 A218012 A218013 * A218015 A218016 A218017

Keyword

nonn

Author

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

Status

approved