|
| |
|
|
A117330
|
|
a(n) = 3 X 3 determinant of 9 consecutive primes starting with the n-th prime.
|
|
6
| |
|
|
-78, 20, -36, 36, -40, -96, 96, -480, -424, 520, 348, 100, -540, 144, -144, -712, 240, 96, 480, -1120, -468, -1152, -3384, 1404, -576, -3924, 7884, -1548, -7312, 6288, -1828, -528, -768, 1920, 720, 768, -1920, 2400, -944, -9340, 12588, 15540, -864, 5600, 4124, -13668, -1428, 1552
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| 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 determinat D = aei+bfg+cdh-afh-bdi-ceg. Continuous prime blocks of 9 consecutive primes are substituted into a,b,c,d,e,f,g,h,i to evaluate D.
The first term -78 is 6 mod 12 but all subsequent terms are 0,4,8 mod 12. Checked out to n=10000. A117329 is the subsequence formed by taking every 9th term.
The smallest absolute value of the sequence is 0.
|
|
|
EXAMPLE
| The second block of 9-primes in continuous fashion is 3,5,7,11,13,17,19,23,29.
So D = a*e*i+b*f*g+c*d*h-a*f*h-b*d*i-c*e*g = 3*13*29+5*17*19+7*11*23-3*17*23-5*11*29-7*13*19 = 20, the second term in the sequence.
a(3)=-36 since det([[5,7,11],[13,17,19],[23,29,31]])=-36.
|
|
|
MAPLE
| primedet := proc(n) local L; L:=map(ithprime, [$n..n+8]); linalg[det]([L[1..3], L[4..6], L[7..9]]) end;
|
|
|
PROG
| (PARI) det3cont(n) = \ determinants of 3 X 3 continuous prime matrices { local(a, b, c, d, e, f, g, h, i, m=0, p=0, x, D); for(x=1, n, 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++); \ if(D==0, print(x", "prime(x)", ")); print1(D", "); ); print(); print("neg= "m); print("pos= "p); print("pos/neg = "p/m+.) }
|
|
|
CROSSREFS
| Cf. A117329.
Sequence in context: A196543 A201983 A098024 * A033398 A204376 A176094
Adjacent sequences: A117327 A117328 A117329 * A117331 A117332 A117333
|
|
|
KEYWORD
| easy,sign
|
|
|
AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com) and Walter Kehowski (wkehowski(AT)cox.net), Apr 24 2006
|
|
|
EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Stefan Steinerberger, Jul 14 2007
|
| |
|
|
