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

Author

Roger L. Bagula, Nov 13 2008

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

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 *)

Ruby

require 'mathn'
Prime.take(100).each_cons(3).select{ |a,b,c| b-a>c-b }.map(&:first)
-- Aaron Weiner, Dec 05 2013

Extensions

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

Simpler formula added by Aaron Weiner, Dec 05 2013