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

1,1

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

Table of n, a(n) for n=1..51.

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

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