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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
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.
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(n=9*40, 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, ", "); )
}
CROSSREFS
Sequence in context: A003913 A251321 A074089 * A057798 A057800 A057806
KEYWORD
sign
AUTHOR
Cino Hilliard, Apr 24 2006
EXTENSIONS
Corrected and extended by Harvey P. Dale, Mar 24 2013
STATUS
approved