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A131175
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Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.
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1
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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
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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A_1 = [2], with determinant = 2 and characteristic polynomial = x-2, 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.
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MAPLE
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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
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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