A117027 Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes.

-1, -12, -48, 152, 102, -60, -24, -72, -24, -60, -24, 1942, 896, -1124, 522, -1938, 554, 1376, -806, -96, -24, 1716, 4598, -48, 2886, -2348, -96, 2380, 4908, 1246, -4158, -180, -1526, 1442, -1882, -4986, -5214, -72, -96, -5694, -2014, -84, -2058, 8232, -4324, -24, -84, -14076, -4844, -7398, 12274

Offset

1,2

Comments

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.

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:

n ratio pos/neg first difference of ratio:

10^2 0.66666666666...

10^3 0.61290322580... -0.053

10^4 0.76928520877... 0.156

10^5 0.83163601729... 0.062

10^6 0.85985612153... 0.028

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

Links

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

Formula

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

Example

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

Maple

A117027 := proc(n) ithprime(4*n-3)*ithprime(4*n)-ithprime(4*n-2)*ithprime(4*n-1) ; end proc:
seq(A117027(n), n=1..30) ; # R. J. Mathar, Mar 27 2010

Prog

(PARI) det(n) = {local(m, p, x, D); m=0; p=0;
forstep(x=1, n, 4, D=prime(x)*prime(x+3)-prime(x+1)*prime(x+2); if(D<0, m++, p++); print1(D", ") );
print(); print("neg= "m); print("pos= "p); print("pos/neg = "p/m+.) }

Crossrefs

Sequence in context: A059162 A333032 A190622 * A161171 A007200 A061148

Adjacent sequences: A117024 A117025 A117026 * A117028 A117029 A117030

Keyword

sign,easy

Author

Cino Hilliard, Apr 24 2006

Extensions

Formula reformulated using standard nomenclature by R. J. Mathar, Mar 27 2010

Status

approved