Search: id:A117330
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%I A117330
%S A117330 78,20,36,36,40,96,96,480,424,520,348,100,540,144,144,712,240,96,480,
%T A117330 1120,468,1152,3384,1404,576,3924,7884,1548,7312,6288,1828,528,768,1920,
%U A117330 720,768,1920,2400,944,9340,12588,15540,864,5600,4124,13668,1428,1552
%V A117330 -78,20,-36,36,-40,-96,96,-480,-424,520,348,100,-540,144,-144,-712,240,
               96,480,-1120,
%W A117330 -468,-1152,-3384,1404,-576,-3924,7884,-1548,-7312,6288,-1828,-528,-768,
               1920,720,768,
%X A117330 -1920,2400,-944,-9340,12588,15540,-864,5600,4124,-13668,-1428,1552
%N A117330 a(n) = 3 X 3 determinant of 9 consecutive primes starting with the n-th 
               prime.
%C A117330 A 3 X 3 matrix with elements of first row a,b,c and second row d,e,f 
               and third row g,h,i has a determinat D = aei+bfg+cdh-afh-bdi-ceg. 
               Continuous prime blocks of 9 consecutive primes are substituted into 
               a,b,c,d,e,f,g,h,i to evaluate D.
%C A117330 The first term -78 is 6 mod 12 but all subsequent terms are 0,4,8 mod 
               12. Checked out to n=10000. A117329 is the subsequence formed by 
               taking every 9th term.
%C A117330 The smallest absolute value of the sequence is 0.
%e A117330 The second block of 9-primes in continuous fashion is 3,5,7,11,13,17,
               19,23,29.
%e A117330 So D = a*e*i+b*f*g+c*d*h-a*f*h-b*d*i-c*e*g = 3*13*29+5*17*19+7*11*23-3*17*23-5*11*29-7*13*19 
               = 20, the second term in the sequence.
%e A117330 a(3)=-36 since det([[5,7,11],[13,17,19],[23,29,31]])=-36.
%p A117330 primedet := proc(n) local L; L:=map(ithprime,[$n..n+8]); linalg[det]([L[1..3],
               L[4..6],L[7..9]]) end;
%o A117330 (PARI) det3cont(n) = \ determinants of 3 X 3 continuous prime matrices 
               { local(a,b,c,d,e,f,g,h,i,m=0,p=0,x,D); for(x=1,n, a=prime(x); b=prime(x+1); 
               c=prime(x+2); d=prime(x+3); e=prime(x+4); f=prime(x+5); g=prime(x+6); 
               h=prime(x+7); i=prime(x+8); D = a*e*i+b*f*g+c*d*h-a*f*h-b*d*i-c*e*g; 
               if(D<0,m++,p++); \ if(D==0,print(x","prime(x)",")); print1(D","); 
               ); print(); print("neg= "m); print("pos= "p); print("pos/neg = "p/
               m+.) }
%Y A117330 Cf. A117329.
%Y A117330 Sequence in context: A115616 A120799 A098024 this_sequence A033398 A124289 
               A053083
%Y A117330 Adjacent sequences: A117327 A117328 A117329 this_sequence A117331 A117332 
               A117333
%K A117330 easy,sign
%O A117330 1,1
%A A117330 Cino Hilliard (hillcino368(AT)gmail.com) and Walter Kehowski (wkehowski(AT)cox.net), 
               Apr 24 2006
%E A117330 Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion 
               of Stefan Steinerberger, Jul 14 2007

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