cp's OEIS Frontend

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

%I A117329 #17 Aug 29 2024 02:09:51
%S A117329 -78,520,480,-1548,-1920,-13668,1408,-1316,-1252,11760,12264,16992,
%T A117329 14520,16220,-144,-87960,31428,35340,-1008,-1008,240,43464,-84768,264,
%U A117329 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
%N A117329 Determinants of 3 X 3 matrices of discrete blocks of 9 consecutive primes.
%C A117329 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.
%F A117329 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.
%e A117329 The first block of 9-primes is 2,3,5,7,11,13,17,19,23. So
%e A117329 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.
%t A117329 Table[Det[Partition[Prime[Range[9n+1,9n+9]],3]],{n,0,50}] (* _Harvey P. Dale_, Mar 24 2013 *)
%o A117329 (PARI) det3(n) = \\ determinants of 3 X 3 discrete prime matrices
%o A117329 { 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, ", ");)
%o A117329 }
%K A117329 sign
%O A117329 1,1
%A A117329 _Cino Hilliard_, Apr 24 2006
%E A117329 Corrected and extended by _Harvey P. Dale_, Mar 24 2013