A117329 Determinants of 3 X 3 matrices of discrete blocks of 9 consecutive primes.

-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

Author

Cino Hilliard, Apr 24 2006

Keywords

• sign

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.

Examples

			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.
		

Programs

Mathematica

Table[Det[Partition[Prime[Range[9n+1,9n+9]],3]],{n,0,50}] (* Harvey P. Dale, Mar 24 2013 *)

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, ", ");)
}

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.

Extensions

Corrected and extended by Harvey P. Dale, Mar 24 2013