A147812 Primes prime(n) such that prime(n+1) - prime(n) > prime(n+2) - prime(n+1).

7, 13, 23, 31, 37, 53, 61, 67, 73, 89, 97, 103, 113, 131, 139, 157, 173, 181, 193, 211, 223, 233, 241, 263, 271, 277, 293, 307, 317, 337, 359, 373, 389, 409, 421, 433, 449, 457, 467, 479, 491, 509, 523

Offset

1,1

Comments

This was originally formulated as (-prime(n) + 2*prime(n+1) - prime(n+2))/((1 - prime(n) + prime(n+1))^(3/2)) > 0, which relates it to other sequences. This is equivalent since the denominator is always positive.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Example

The gap between 7 and the next prime, 11, is 4, which is greater than the next prime gap from 11 to 13, so 7 is in the sequence.

Mathematica

d2[n_] = Prime[n + 2] - 2*Prime[n + 1] + Prime[n]; d1[n_] = Prime[n + 1] - Prime[n]; k[n_] = -d2[n]/(1 + d1[n])^(3/2); Flatten[Table[If[k[n] > 0, Prime[n], {}], {n, 1, 100}]]
Select[Partition[Prime[Range[150]], 3, 1], #[[2]]-#[[1]]>#[[3]]-#[[2]]&][[All, 1]] (* Harvey P. Dale, Mar 29 2022 *)

Prog

(Haskell)
import Data.List (findIndices)
a147812 n = a147812_list !! (n-1)
a147812_list = map (a000040 . (+ 1)) $ findIndices (< 0) a036263_list
-- Reinhard Zumkeller, Jan 20 2012
(Ruby)
require 'mathn'
Prime.take(100).each_cons(3).select{ |a, b, c| b-a>c-b }.map(&:first)
-- Aaron Weiner, Dec 05 2013

Crossrefs

Cf. A036263, A147813 (complement with respect to A000040).

Sequence in context: A106349 A293657 A048449 * A043884 A129727 A275897

Adjacent sequences: A147809 A147810 A147811 * A147813 A147814 A147815

Keyword

nonn

Author

Roger L. Bagula, Nov 13 2008

Extensions

Edited by Alonso del Arte and Joerg Arndt, Nov 01 2013

Simpler formula added by Aaron Weiner, Dec 05 2013

Status

approved