

A131175


Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.


1



1, 2, 1, 8, 1, 26, 4, 1, 66, 36, 1, 174, 196, 1, 398, 676, 1, 878, 3044, 1, 2174, 6852, 192, 1, 4862, 18628, 704, 1, 10494, 45508, 1216, 1, 22014, 141252, 6336, 1, 47614, 315332, 10432, 1, 100862, 858052, 55488, 1, 225278, 1878980, 245952
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OFFSET

1,2


COMMENTS

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] = nth nalmost prime A101695. The second column of the table is the negative of the trace of the matrices.


LINKS



FORMULA

Row n of the table consists of the coefficients of x^n, x^n1, ... of the characteristic polynomial of the n X n matrix A whose first row is the first n primes (1almost primes) (A000040), 2nd row is the first n semiprimes (2almost primes) A001358, 3rd row is the first n 3almost primes A014612.


EXAMPLE

A_1 = [2], with determinant = 2 and characteristic polynomial = x2, with coefficients (1, 2) so a(a) = 1 and a(2) = 2.
A_2 =
[2.3]
[4.6]
with determinant = 0, polynomial x^2  8x, so the coefficients are (1, 8), hence a(3) = 1 and a(4) = 8.
A_3 =
[2..3..5]
[4..6..9]
[8.12.18]
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.


MAPLE

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] := xA078840(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


CROSSREFS



KEYWORD

sign,tabf


AUTHOR



EXTENSIONS



STATUS

approved



