1,1

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.

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.

The smallest absolute value of the sequence is 0.

The second block of 9-primes in continuous fashion is 3,5,7,11,13,17,19,23,29.

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.

a(3)=-36 since det([[5,7,11],[13,17,19],[23,29,31]])=-36.

primedet := proc(n) local L; L:=map(ithprime, [$n..n+8]); linalg[det]([L[1..3], L[4..6], L[7..9]]) end;

(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+.) }

Cf. A117329.

Adjacent sequences: A117327 A117328 A117329 this_sequence A117331 A117332 A117333

Sequence in context: A115616 A120799 A098024 this_sequence A033398 A124289 A053083

easy,sign

Cino Hilliard (hillcino368(AT)gmail.com) and Walter Kehowski (wkehowski(AT)cox.net), Apr 24 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Stefan Steinerberger, Jul 14 2007