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

%I #12 Nov 18 2022 18:08:38

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

%C 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 Harvey P. Dale, <a href="/A136612/b136612.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n)=A001223(n)+A001223(n+2). - _R. J. Mathar_, Apr 21 2008

%e 2 + 5 = 7

%e 3 + 7 = 10

%e 5 + 11 = 16

%e 7 + 13 = 20

%e ...

%e so the sequence is: 10 - 7 = 3,

%e 16 - 10 = 6,

%e 20 - 16 = 4,

%e 28 - 20 = 8,

%e ...

%p 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 #[[4]]+#[[2]]-#[[3]]-#[[1]]&/@Partition[Prime[Range[90]],4,1] (* _Harvey P. Dale_, May 15 2013 *)

%o (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 Cf. A000040, A001223.

%K easy,nonn

%O 1,1

%A _Odimar Fabeny_, Apr 14 2008

%E Corrected and extended by _R. J. Mathar_, Apr 21 2008