A199514 Numerators of zeros to a symmetric polynomial. Numerators of mu(n)^2*(n/(n - phi(n))).

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 an integer only for prime numbers. As the matrix gets bigger there are fractions as zeros that are greater than small prime numbers.

Links

Antti Karttunen, Table of n, a(n) for n = 2..65537

Mats Granvik, Are the primes found as a subset in this sequence?, Mathematics Stack Exchange.

Formula

a(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 7 X 7 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

Adjacent sequences: A199511 A199512 A199513 * A199515 A199516 A199517

Keyword

nonn,frac,easy,changed

Author

Mats Granvik, Nov 07 2011

Status

approved