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A136612 a(n) = ((prime(n+3) + prime(n+1)) - (prime(n+2) + prime(n))).

%I A136612 #12 Nov 18 2022 18:08:38
%S A136612 3,6,4,8,4,8,8,6,12,6,8,8,8,10,8,12,6,8,10,6,12,12,10,10,8,4,8,16,8,
%T A136612 20,6,16,4,16,8,10,12,10,8,16,4,14,4,16,14,16,14,8,8,6,16,8,16,12,8,
%U A136612 12,6,8,14,16,14,16,8,16,10,24,8,14,8,12,12,14,10,12,12,10,16,14,10,20,4,16,6
%N A136612 a(n) = ((prime(n+3) + prime(n+1)) - (prime(n+2) + prime(n))).
%C A136612 a(n) is the sum of two prime gaps, thus a(n) >= 4 for n > 1. Conjecturally a(n) << log^2 n (probably with constant around 2). - _Charles R Greathouse IV_, Aug 25 2014
%H A136612 Harvey P. Dale, <a href="/A136612/b136612.txt">Table of n, a(n) for n = 1..1000</a>
%F A136612 a(n)=A001223(n)+A001223(n+2). - _R. J. Mathar_, Apr 21 2008
%e A136612 2 + 5 = 7
%e A136612 3 + 7 = 10
%e A136612 5 + 11 = 16
%e A136612 7 + 13 = 20
%e A136612 ...
%e A136612 so the sequence is: 10 - 7 = 3,
%e A136612 16 - 10 = 6,
%e A136612 20 - 16 = 4,
%e A136612 28 - 20 = 8,
%e A136612 ...
%p A136612 A001223 := proc(n) ithprime(n+1)-ithprime(n) ; end: A136612 := proc(n) A001223(n)+A001223(n+2) ; end: seq(A136612(n),n=1..100) ; # _R. J. Mathar_, Apr 21 2008
%t A136612 #[[4]]+#[[2]]-#[[3]]-#[[1]]&/@Partition[Prime[Range[90]],4,1] (* _Harvey P. Dale_, May 15 2013 *)
%o A136612 (PARI) a(n)=my(p=prime(n),q=nextprime(p+1),r=nextprime(q+1)); nextprime(r+1)-r + q-p \\ _Charles R Greathouse IV_, Aug 25 2014
%Y A136612 Cf. A000040, A001223.
%K A136612 easy,nonn
%O A136612 1,1
%A A136612 _Odimar Fabeny_, Apr 14 2008
%E A136612 Corrected and extended by _R. J. Mathar_, Apr 21 2008