A316917 Let g(n) be the n-th maximal prime gap; a(n) = 1 if g(n) has record merit, 0 if it does not.

1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Author

Rodolfo Ruiz-Huidobro, Jul 16 2018

Keywords

• nonn
• more

Comments

a(n) = 1 if A002386(n) is in A111870, a(n) = 0 if A002386(n) is not in A111870.

The merit M of a prime gap of measure g following the prime p_1 is defined as M=g/ln(p_1). It is the ratio of the measure of the gap to the "average" measure of gaps near that point. As an example, the merit of the sixth maximal gap, of size 14, after prime 113 is 2.96.

a(81) = 0 because there are previous maximal gaps with higher merits. - Rodolfo Ruiz-Huidobro, Jan 23 2024

a(82) = 1 as the merit of the gap is 1572/log(18571673432051830099)=1572/44.37=35.43 (which is a record merit). - Rodolfo Ruiz-Huidobro, May 10 2024

a(83) =1 as the merit for the gap is 1676/log(20733746510561444539) =1676/44.48=37.681 (which is a record merit). - Rodolfo Ruiz-Huidobro, Dec 20 2024

Examples

			The 5th record prime gap from 89 to 97 does not have record merit, so a(5) = 0.
The 10th record prime gap from 1327 to 1361 has record merit, so a(10) = 1.
		

Links

• Prime Gap List Community, Record prime gaps, 2021.
• Index entries for characteristic functions

Crossrefs

Cf. A000101, A002386, A111870, A111871, A005250.

Programs

Mathematica

Block[{nn = 10^6, s, t, u, v}, s = Prime@ Range[nn]; t = Differences@ s; u = Map[(#2 - #1)/Log[#1] & @@ # &, Partition[Prime@ Range[nn], 2, 1]]; v = Map[Prime@ FirstPosition[u, #][[1]] &, Union@ FoldList[Max, u]]; Boole[! FreeQ[v, s[[FirstPosition[t, #][[1]] ]] ] ] & /@ Union@ FoldList[Max, t]] (* Michael De Vlieger, Jul 19 2018 *)

Extensions

a(81) from Rodolfo Ruiz-Huidobro, Jan 23 2024

a(82) from Rodolfo Ruiz-Huidobro, May 10 2024

a(83) from Rodolfo Ruiz-Huidobro, Dec 09 2024