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 A117329 Determinants of 3 X 3 matrices of discrete blocks of 9 consecutive primes. 1
 -78, 520, 480, -1548, -1920, -13668, 1408, -1316, -1252, 11760, 12264, 16992, 14520, 16220, -144, -87960, 31428, 35340, -1008, -1008, 240, 43464, -84768, 264, 431340, 45824, -28540, -29484, -56916, -672, 120960, -54260, 18164, 31528, -101736, -258264, 356448, 73440, 149552, -18616, 117864, 12620, 125280, 22064, -55428, 112272, -4992, -214536, -72184, 885960, 333720 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS FORMULA 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 determinant 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. EXAMPLE 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. MATHEMATICA Table[Det[Partition[Prime[Range[9n+1, 9n+9]], 3]], {n, 0, 50}] (* Harvey P. Dale, Mar 24 2013 *) PROG (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+.) } CROSSREFS Sequence in context: A003913 A251321 A074089 * A057798 A057800 A057806 Adjacent sequences:  A117326 A117327 A117328 * A117330 A117331 A117332 KEYWORD sign AUTHOR Cino Hilliard, Apr 24 2006 EXTENSIONS Corrected and extended by Harvey P. Dale, Mar 24 2013 STATUS approved

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Last modified September 22 22:38 EDT 2021. Contains 347609 sequences. (Running on oeis4.)