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A199514
Numerators of zeros to a symmetric polynomial. Numerators of mu(n)^2*(n/(n - phi(n))).
4
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
OFFSET
2,1
COMMENTS
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.
LINKS
Mats Granvik, Are the primes found as a subset in this sequence?, Mathematics Stack Exchange.
FORMULA
A199514(n)/A199515(n) = A008683(n)^2*(n/(n - A000010(n))), n > 1.
a(n) = numerator of A008966(n)*(n/A051953(n)). - Antti Karttunen, Sep 07 2018
EXAMPLE
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
...
MATHEMATICA
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]
PROG
(PARI) A199514(n) = numerator(issquarefree(n)*(n/(n-eulerphi(n)))); \\ Antti Karttunen, Sep 07 2018
CROSSREFS
Cf. A000010, A008683, A008966, A051953, A191898. Denominators: A199515.
Sequence in context: A080368 A057174 A197658 * A080367 A354365 A066913
KEYWORD
nonn,frac,easy
AUTHOR
Mats Granvik, Nov 07 2011
STATUS
approved