A147812 - OEIS

%I #28 Mar 29 2022 19:29:14

%S 7,13,23,31,37,53,61,67,73,89,97,103,113,131,139,157,173,181,193,211,

%T 223,233,241,263,271,277,293,307,317,337,359,373,389,409,421,433,449,

%U 457,467,479,491,509,523

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

%C 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.

%H Reinhard Zumkeller, <a href="/A147812/b147812.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%t 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}]]

%t Select[Partition[Prime[Range[150]],3,1],#[[2]]-#[[1]]>#[[3]]-#[[2]]&][[All,1]] (* _Harvey P. Dale_, Mar 29 2022 *)

%o (Haskell)

%o import Data.List (findIndices)

%o a147812 n = a147812_list !! (n-1)

%o a147812_list = map (a000040 . (+ 1)) $ findIndices (< 0) a036263_list

%o -- _Reinhard Zumkeller_, Jan 20 2012

%o (Ruby)

%o require 'mathn'

%o Prime.take(100).each_cons(3).select{ |a,b,c| b-a>c-b }.map(&:first)

%o -- _Aaron Weiner_, Dec 05 2013

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

%K nonn

%O 1,1

%A _Roger L. Bagula_, Nov 13 2008

%E Edited by _Alonso del Arte_ and _Joerg Arndt_, Nov 01 2013

%E Simpler formula added by _Aaron Weiner_, Dec 05 2013