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

%I A199514 #30 Jan 14 2025 13:51:43
%S A199514 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 A199514 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 A199514 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 A199514 Numerators of zeros to a symmetric polynomial. Numerators of mu(n)^2*(n/(n - phi(n))).
%C A199514 The polynomials are defined as the determinant of a symmetric matrix with the following definition:
%C A199514 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 A199514 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 A199514 Antti Karttunen, <a href="/A199514/b199514.txt">Table of n, a(n) for n = 2..65537</a>
%H A199514 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 A199514 a(n)/A199515(n) = A008683(n)^2*(n/(n - A000010(n))), n > 1.
%F A199514 a(n) = numerator of A008966(n)*(n/A051953(n)). - _Antti Karttunen_, Sep 07 2018
%e A199514 The 7 X 7 symmetric matrix is:
%e A199514   1......1......1......1......1......1......1
%e A199514   1...-1+x......1...-1+x......1...-1+x......1
%e A199514   1......1...-2+x......1......1...-2+x......1
%e A199514   1...-1+x......1.....-1......1...-1+x......1
%e A199514   1......1......1......1...-4+x......1......1
%e A199514   1...-1+x...-2+x...-1+x......1...2-2x......1
%e A199514   1......1......1......1......1......1...-6+x
%e A199514 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 A199514 The polynomials for the first seven matrices are:
%e A199514   1,
%e A199514   -2 + x,
%e A199514   6 - 5 x + x^2,
%e A199514   -6 x + 5 x^2 - x^3,
%e A199514   30 x - 31 x^2 + 10 x^3 - x^4,
%e A199514   180 x - 306 x^2 + 184 x^3 - 46 x^4 + 4 x^5,
%e A199514   -1260 x + 2322 x^2 - 1594 x^3 + 506 x^4 - 74 x^5 + 4 x^6,
%e A199514   ...
%e A199514 and their zeros respectively are:
%e A199514   {}
%e A199514   2
%e A199514   2, 3
%e A199514   2, 3, 0
%e A199514   2, 3, 0, 5
%e A199514   2, 3, 0, 5, 3/2
%e A199514   2, 3, 0, 5, 3/2, 7
%e A199514   ...
%t A199514 Table[Numerator[MoebiusMu[n]^2*(n/(n - EulerPhi[n]))], {n, 2, 90}]
%t A199514 (* or *)
%t A199514 Clear[nn, t, n, k, M, x];
%t A199514 nn = 90;
%t A199514 a = Range[nn]*0;
%t A199514 Do[
%t A199514 t[n_, 1] = 1;
%t A199514 t[1, k_] = 1;
%t A199514 t[n_, k_] :=
%t A199514   t[n, k] =
%t A199514    If[n < k,
%t A199514     If[And[n > 1, k > 1], x - Sum[t[k - i, n], {i, 1, n - 1}], 0],
%t A199514     If[And[n > 1, k > 1], x - Sum[t[n - i, k], {i, 1, k - 1}], 0]];
%t A199514 M = Table[Table[t[n, k], {k, 1, i}], {n, 1, i}];
%t A199514 a[[i]] = x /. Solve[Det[M] == 0, x], {i, 1, nn}];
%t A199514 a[[1]] = {};
%t A199514 b = Differences[Table[Total[a[[i]]], {i, 1, nn}]];
%t A199514 Numerator[b]
%o A199514 (PARI) A199514(n) = numerator(issquarefree(n)*(n/(n-eulerphi(n)))); \\ _Antti Karttunen_, Sep 07 2018
%Y A199514 Cf. A000010, A008683, A008966, A051953, A191898. Denominators: A199515.
%K A199514 nonn,frac,easy
%O A199514 2,1
%A A199514 _Mats Granvik_, Nov 07 2011