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Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes.
1

%I #11 Aug 09 2015 14:18:22

%S -1,-12,-48,152,102,-60,-24,-72,-24,-60,-24,1942,896,-1124,522,-1938,

%T 554,1376,-806,-96,-24,1716,4598,-48,2886,-2348,-96,2380,4908,1246,

%U -4158,-180,-1526,1442,-1882,-4986,-5214,-72,-96,-5694,-2014,-84,-2058,8232,-4324,-24,-84,-14076,-4844,-7398,12274

%N Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes.

%C The terminology "non-overlapping" is used to distinguish from "continuous" which would be block 1: 2, 3, 5, 7; block 2: 3, 5, 7, 11 etc.

%C The number of negative values in this sequence appears to be consistently larger than the number of positive values. For n=10^x we have ratios:

%C n ratio pos/neg first difference of ratio:

%C 10^2 0.66666666666...

%C 10^3 0.61290322580... -0.053

%C 10^4 0.76928520877... 0.156

%C 10^5 0.83163601729... 0.062

%C 10^6 0.85985612153... 0.028

%C This suggests the ratio is approaching a limit close to 0.87.

%F a(n) = prime(4n-3)*prime(4n) - prime(4n-2)*prime(4n-1).

%e The 4th block of 4-primes is 41,43,47,53. So the determinant is = 41*53 - 43*47 = 152 = a(4).

%p A117027 := proc(n) ithprime(4*n-3)*ithprime(4*n)-ithprime(4*n-2)*ithprime(4*n-1) ; end proc:

%p seq(A117027(n),n=1..30) ; # _R. J. Mathar_, Mar 27 2010

%o (PARI) det(n) = {local(m,p,x,D); m=0; p=0;

%o forstep(x=1,n,4, D=prime(x)*prime(x+3)-prime(x+1)*prime(x+2); if(D<0,m++,p++); print1(D",") );

%o print(); print("neg= "m); print("pos= "p); print("pos/neg = "p/m+.) }

%K sign,easy

%O 1,2

%A _Cino Hilliard_, Apr 24 2006

%E Formula reformulated using standard nomenclature by _R. J. Mathar_, Mar 27 2010