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

%I A147812 #28 Mar 29 2022 19:29:14
%S A147812 7,13,23,31,37,53,61,67,73,89,97,103,113,131,139,157,173,181,193,211,
%T A147812 223,233,241,263,271,277,293,307,317,337,359,373,389,409,421,433,449,
%U A147812 457,467,479,491,509,523
%N A147812 Primes prime(n) such that prime(n+1) - prime(n) > prime(n+2) - prime(n+1).
%C A147812 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 A147812 Reinhard Zumkeller, <a href="/A147812/b147812.txt">Table of n, a(n) for n = 1..10000</a>
%e A147812 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 A147812 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 A147812 Select[Partition[Prime[Range[150]],3,1],#[[2]]-#[[1]]>#[[3]]-#[[2]]&][[All,1]] (* _Harvey P. Dale_, Mar 29 2022 *)
%o A147812 (Haskell)
%o A147812 import Data.List (findIndices)
%o A147812 a147812 n = a147812_list !! (n-1)
%o A147812 a147812_list = map (a000040 . (+ 1)) $ findIndices (< 0) a036263_list
%o A147812 -- _Reinhard Zumkeller_, Jan 20 2012
%o A147812 (Ruby)
%o A147812 require 'mathn'
%o A147812 Prime.take(100).each_cons(3).select{ |a,b,c| b-a>c-b }.map(&:first)
%o A147812 -- _Aaron Weiner_, Dec 05 2013
%Y A147812 Cf. A036263, A147813 (complement with respect to A000040).
%K A147812 nonn
%O A147812 1,1
%A A147812 _Roger L. Bagula_, Nov 13 2008
%E A147812 Edited by _Alonso del Arte_ and _Joerg Arndt_, Nov 01 2013
%E A147812 Simpler formula added by _Aaron Weiner_, Dec 05 2013