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%I #14 May 25 2018 08:30:52
%S 4,13,14,30,31,67,68,77,78,127,128,129,138,139,188,197,201,217,221,
%T 222,238,247,263,288,297,322,331,332,333,349,353,354,355,476,501,517,
%U 526,527,531,532,533,569,585,586,635,636,637,641,642,723,732,733,737,762
%N Sum of squares of gaps between consecutive semiprimes.
%C This is to semiprimes A001358 as A074741 is to primes A000040. What is the semiprime analog of D. R. Heath-Brown's conjecture: Sum_{prime(n)<=N} (prime(n)-prime(n-1))^2 ~ 2*N*log(N) and Marek Wolf's conjecture: Sum_{prime(n)<N} (prime(n)-prime(n-1))^2 = 2*N^2/pi(N) + error term(N), pi(N)=A000720(n).
%H Harvey P. Dale, <a href="/A135406/b135406.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = SUM[k=1..n] A065516(k)^2 = SUM[k=1..n] (A001358(n+1) - A001358(n))^2.
%e a(10) = (6-4)^2 + (9-6)^2 + (10-9)^2 + (14-10)^2 + (15-14)^2 + (21-15)^2 + (22-21)^2 + (25-22)^2 + (26-25)^2 + (33-26)^2 = (2^2) + (3^2) + (1^2) + (4^2) + (1^2) + (6^2) + (1^2) + (3^2) + (1^2) + (7^2) = 127.
%p A001358 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A065516 := proc(n) A001358(n+1)-A001358(n) ; end: A135406 := proc(n) add( (A065516(k))^2,k=1..n) ; end: seq(A135406(n),n=1..80) ; # _R. J. Mathar_, Jan 07 2008
%t Accumulate[Differences[Select[Range[200],PrimeOmega[#]==2&]]^2] (* _Harvey P. Dale_, Mar 05 2015 *)
%Y Cf. A000040, A001358, A065516, A074741.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Dec 09 2007
%E More terms from _R. J. Mathar_, Jan 07 2008
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