|
%I #11 Aug 24 2015 02:27:52
%S 1,-2,1,-8,1,-26,-4,1,-66,-36,1,-174,-196,1,-398,-676,1,-878,-3044,1,
%T -2174,-6852,-192,1,-4862,-18628,-704,1,-10494,-45508,-1216,1,-22014,
%U -141252,-6336,1,-47614,-315332,-10432,1,-100862,-858052,-55488,1,-225278,-1878980,-245952
%N Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.
%C Because the first column of A is a column vector of powers of 2, the determinant (for n>1) is always 0. Hence the rank is always (for n>1) less than n. A[n.n] = n-th n-almost prime A101695. The second column of the table is the negative of the trace of the matrices.
%F Row n of the table consists of the coefficients of x^n, x^n-1, ... of the characteristic polynomial of the n X n matrix A whose first row is the first n primes (1-almost primes) (A000040), 2nd row is the first n semiprimes (2-almost primes) A001358, 3rd row is the first n 3-almost primes A014612.
%e A_1 = [2], with determinant = 2 and characteristic polynomial = x-2, with coefficients (1, -2) so a(a) = 1 and a(2) = -2.
%e A_2 =
%e [2.3]
%e [4.6]
%e with determinant = 0, polynomial x^2 - 8x, so the coefficients are (1, -8), hence a(3) = 1 and a(4) = -8.
%e A_3 =
%e [2..3..5]
%e [4..6..9]
%e [8.12.18]
%e with determinant = 0, polynomial = x^3 - 26x^2, -4x, so coefficients are (1, -26, -4), hence a(5) = 1, a(6) = -26, a(7) = -4.
%p A078840 := proc(n,m) local p,k ; k := 1 ; p := 2^n ; while k < m do p := p+1 ; while numtheory[bigomega](p) <> n do p := p+1 ; od; k := k+1 ; od: RETURN(p) ; end: A131175 := proc(nrow,showall) local A,row,col,pol,T,a ; A := linalg[matrix](nrow,nrow) ; for row from 1 to nrow do for col from 1 to nrow do if row = col then A[row,col] := x-A078840(row,col) ; else A[row,col] := -A078840(row,col) ; fi ; od: od: pol := linalg[det](A) ; T := [] ; for col from nrow to 0 by -1 do a := coeftayl(pol,x=0,col) ; if a <> 0 or showall then T := [op(T),a] ; fi ; od; RETURN(T) ; end: for n from 1 to 15 do print(op(A131175(n,false))) ; od: # _R. J. Mathar_, Oct 26 2007
%Y Cf. A000040, A001358, A014612, A014613, A014614, A101695.
%K sign,tabf
%O 1,2
%A _Jonathan Vos Post_, Sep 24 2007
%E Corrected and extended by _R. J. Mathar_, Oct 26 2007
|
