1,1

The number of negative values in this sequence appears to become smaller and smaller than the number of positive values. This suggests the ratios of these two numbers approach a limit as the number of terms increases. The smallest absolute value of the determinants in this sequence is 0. For example x=1009 in the PARI script will give a determinant of 0.

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. Discrete prime blocks of 9 consecutive primes are substituted into a,b,c,d,e,f,g,h,i to evaluate D.

The first block of 9-primes is 2,3,5,7,11,13,17,19,23. So

D = 2*11*23+3*13*17+5*7*19-2*13*19-3*7*23-5*11*17 = -78, the first entry in the table.

(PARI) det3(n) = \ determinants of 3 X 3 discrete prime matrices { local(a, b, c, d, e, f, g, h, i, m=0, p=0, x, D); forstep(x=1, n, 9, 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++); print1(D", "); ); print(); print("neg= "m); print("pos= "p); print("pos/neg = "p/m+.) }

Sequence in context: A007255 A003913 A074089 this_sequence A057798 A057800 A057806

Adjacent sequences: A117326 A117327 A117328 this_sequence A117330 A117331 A117332

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Cino Hilliard (hillcino368(AT)gmail.com), Apr 24 2006