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Search: id:A117027
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| A117027 |
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Determinants of 2 X 2 matrices of discrete blocks of 4 consecutive primes. |
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+0
1
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| -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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The terminology "discrete" is used to distinguish from "continuous" which would be block 1: 2,3,5,7, block2: 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 pos/neg diff
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.
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FORMULA
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A 2 by 2 matrix with elements first row a,b and second row c,d has a determinat D = ad - bc. Discrete prime blocks of 4 are substituted into a,b,c,d to evaluate D.
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EXAMPLE
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The 4th block of 4-primes is 41,43,47,53. So det = 41*53 - 43*47 = 152, the 4th entry in the table.
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PROGRAM
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(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+.) }
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CROSSREFS
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Sequence in context: A135453 A006564 A059162 this_sequence A007200 A061148 A052601
Adjacent sequences: A117024 A117025 A117026 this_sequence A117028 A117029 A117030
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KEYWORD
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sign,uned
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Apr 24 2006
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