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A199514
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Numerators of zeros to a symmetric polynomial. Numerators of mu(n)^2*(n/(n - phi(n))).
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4
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2, 3, 0, 5, 3, 7, 0, 0, 5, 11, 0, 13, 7, 15, 0, 17, 0, 19, 0, 7, 11, 23, 0, 0, 13, 0, 0, 29, 15, 31, 0, 33, 17, 35, 0, 37, 19, 13, 0, 41, 7, 43, 0, 0, 23, 47, 0, 0, 0, 51, 0, 53, 0, 11, 0, 19, 29, 59, 0, 61, 31, 0, 0, 65, 33, 67, 0, 69, 35, 71, 0, 73, 37, 0, 0, 77, 13, 79, 0, 0, 41, 83, 0, 85, 43, 87, 0, 89, 0
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OFFSET
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2,1
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COMMENTS
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The polynomials are defined as the determinant of a symmetric matrix with the following definition:
T(n, 1) = 1, T(1, k) = 1, T(n, k) = If n < k, x - Sum_(i = 1)^(i = n - 1) of T(k - i, n), otherwise x - Sum_(i = 1)^(i = k - 1) of T(k - i, n).
Eric Naslund on Mathematics Stack Exchange kindly gave the description in terms of arithmetic functions. The sequence of fractions A199514/A199515 is integer only for prime numbers. As the matrix gets bigger there are fractions as zeros that are greater than small prime numbers.
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LINKS
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FORMULA
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EXAMPLE
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The seven times seven symmetric matrix is:
1......1......1......1......1......1......1
1...-1+x......1...-1+x......1...-1+x......1
1......1...-2+x......1......1...-2+x......1
1...-1+x......1.....-1......1...-1+x......1
1......1......1......1...-4+x......1......1
1...-1+x...-2+x...-1+x......1...2-2x......1
1......1......1......1......1......1...-6+x
Taking the determinant of the matrix above gives the polynomial: -1260 x + 2322 x^2 - 1594 x^3 + 506 x^4 - 74 x^5 + 4 x^6
The polynomials for the first seven matrices are:
1,
-2 + x,
6 - 5 x + x^2,
-6 x + 5 x^2 - x^3,
30 x - 31 x^2 + 10 x^3 - x^4,
180 x - 306 x^2 + 184 x^3 - 46 x^4 + 4 x^5,
-1260 x + 2322 x^2 - 1594 x^3 + 506 x^4 - 74 x^5 + 4 x^6,
...
and their zeros respectively are:
{}
2
2, 3
2, 3, 0
2, 3, 0, 5
2, 3, 0, 5, 3/2
2, 3, 0, 5, 3/2, 7
...
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MATHEMATICA
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Table[Numerator[MoebiusMu[n]^2*(n/(n - EulerPhi[n]))], {n, 2, 90}]
(* or *)
Clear[nn, t, n, k, M, x];
nn = 90;
a = Range[nn]*0;
Do[
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] :=
t[n, k] =
If[n < k,
If[And[n > 1, k > 1], x - Sum[t[k - i, n], {i, 1, n - 1}], 0],
If[And[n > 1, k > 1], x - Sum[t[n - i, k], {i, 1, k - 1}], 0]];
M = Table[Table[t[n, k], {k, 1, i}], {n, 1, i}];
a[[i]] = x /. Solve[Det[M] == 0, x], {i, 1, nn}];
a[[1]] = {};
b = Differences[Table[Total[a[[i]]], {i, 1, nn}]];
Numerator[b]
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PROG
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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