A199514 - OEIS

%I #30 Jan 14 2025 13:51:43

%S 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,

%T 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,

%U 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

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

%C The polynomials are defined as the determinant of a symmetric matrix with the following definition:

%C 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).

%C 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.

%H Antti Karttunen, <a href="/A199514/b199514.txt">Table of n, a(n) for n = 2..65537</a>

%H Mats Granvik, <a href="http://math.stackexchange.com/questions/64194">Are the primes found as a subset in this sequence?</a>, Mathematics Stack Exchange.

%F a(n)/A199515(n) = A008683(n)^2*(n/(n - A000010(n))), n > 1.

%F a(n) = numerator of A008966(n)*(n/A051953(n)). - _Antti Karttunen_, Sep 07 2018

%e The 7 X 7 symmetric matrix is:

%e   1......1......1......1......1......1......1

%e   1...-1+x......1...-1+x......1...-1+x......1

%e   1......1...-2+x......1......1...-2+x......1

%e   1...-1+x......1.....-1......1...-1+x......1

%e   1......1......1......1...-4+x......1......1

%e   1...-1+x...-2+x...-1+x......1...2-2x......1

%e   1......1......1......1......1......1...-6+x

%e 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

%e The polynomials for the first seven matrices are:

%e   1,

%e   -2 + x,

%e   6 - 5 x + x^2,

%e   -6 x + 5 x^2 - x^3,

%e   30 x - 31 x^2 + 10 x^3 - x^4,

%e   180 x - 306 x^2 + 184 x^3 - 46 x^4 + 4 x^5,

%e   -1260 x + 2322 x^2 - 1594 x^3 + 506 x^4 - 74 x^5 + 4 x^6,

%e   ...

%e and their zeros respectively are:

%e   {}

%e   2

%e   2, 3

%e   2, 3, 0

%e   2, 3, 0, 5

%e   2, 3, 0, 5, 3/2

%e   2, 3, 0, 5, 3/2, 7

%e   ...

%t Table[Numerator[MoebiusMu[n]^2*(n/(n - EulerPhi[n]))], {n, 2, 90}]

%t (* or *)

%t Clear[nn, t, n, k, M, x];

%t nn = 90;

%t a = Range[nn]*0;

%t Do[

%t t[n_, 1] = 1;

%t t[1, k_] = 1;

%t t[n_, k_] :=

%t   t[n, k] =

%t    If[n < k,

%t     If[And[n > 1, k > 1], x - Sum[t[k - i, n], {i, 1, n - 1}], 0],

%t     If[And[n > 1, k > 1], x - Sum[t[n - i, k], {i, 1, k - 1}], 0]];

%t M = Table[Table[t[n, k], {k, 1, i}], {n, 1, i}];

%t a[[i]] = x /. Solve[Det[M] == 0, x], {i, 1, nn}];

%t a[[1]] = {};

%t b = Differences[Table[Total[a[[i]]], {i, 1, nn}]];

%t Numerator[b]

%o (PARI) A199514(n) = numerator(issquarefree(n)*(n/(n-eulerphi(n)))); \\ _Antti Karttunen_, Sep 07 2018

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

%K nonn,frac,easy

%O 2,1

%A _Mats Granvik_, Nov 07 2011