# Equation involving hypergeometric functions

I want to solve this equation but nor NSolve nor Solve are able to do this.

Gamma[1 + 8 (-1 + r) r]/(Gamma[1 + r] Gamma[1 - 9 r + 8 r^2]) - (
Gamma[1 + 14 r] Gamma[
1 + 8 (-2 + r) r] HypergeometricPFQRegularized[{1, -((15 r)/2), -(
r/2)}, {1 + (13 r)/2, 1 - (33 r)/2 + 8 r^2}, 1])/(
Gamma[1 + r/2] Gamma[1 + (15 r)/2])==0


I want to find a solution for $r \in \mathbb{N}$, $r>2$.

NSolve[Gamma[1 + 8 (-1 + r) r]/(
Gamma[1 + r] Gamma[1 - 9 r + 8 r^2]) - (
Gamma[1 + 14 r] Gamma[
1 + 8 (-2 + r) r] HypergeometricPFQRegularized[{1, -((15 r)/2), -(
r/2)}, {1 + (13 r)/2, 1 - (33 r)/2 + 8 r^2}, 1])/(
Gamma[1 + r/2] Gamma[1 + (15 r)/2]), r, Integers]


Any suggestion? I get the following error message:

NSolve::nsmet: This system cannot be solved with the methods available to NSolve.

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I wonder if this equation has no solution for r > 2 (integer or real). If I create a function of the left-hand side of the equation and plot that against r (at least up to r=50), the relationship of Log[-f[r]] vs. r looks pretty linear and I don't see the function heading back to zero (ever).

f[r_] := Gamma[
1 + 8 (-1 + r) r]/(Gamma[1 + r] Gamma[1 - 9 r + 8 r^2]) - (Gamma[
1 + 14 r] Gamma[
1 + 8 (-2 + r) r] HypergeometricPFQRegularized[{1, -((15 r)/
2), -(r/2)}, {1 + (13 r)/2, 1 - (33 r)/2 + 8 r^2},
1])/(Gamma[1 + r/2] Gamma[1 + (15 r)/2])

ListLinePlot[Table[{r, Log[-f[1.*r]]}, {r, 1, 50}]]


with output

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