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

Author

Mats Granvik, Nov 07 2011

Keywords

• nonn
• frac
• easy

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.

Examples

			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
  ...
		

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.

Crossrefs

Cf. A000010, A008683, A008966, A051953, A191898. Denominators: A199515.

Programs

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]

PARI

A199514(n) = numerator(issquarefree(n)*(n/(n-eulerphi(n)))); \\ Antti Karttunen, Sep 07 2018

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