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Set kappa_n := A369990(n) / A369991(n) and theta_n := (kappa_n-kappa_{n+1}) / (kappa_{n-1}-kappa_n). Under the hypothesis that theta_{2m} < theta_{2m+2} < theta_{2*m+3} < theta_{2*m+1} for m=1,2,... (verified for all values known so far), we would obtain 0.27887706136895087 < kappa_{21}' < kappa < kappa_{22}' < 0.27887706136898083, which is sharper than formula (3) below. Here, the transformed sequence (kappa_n') = G(kappa_n) is defined via kappa_n' := (kappa_{n-1}*kappa_{n+1} - kappa_n^2) / (kappa_{n-1} - 2*kappa_n + kappa_{n+1}). (See the first arXiv article for a proof of this conjecture-dependent statement.) Feeling even more adventurous, we could apply the transformation G four times and would obtain 0.278877061368975064775 < kappa_{19}'''' < kappa < kappa_{18}'''' < 0.278877061368975064815.

Roland Miyamoto, <a href="https://arxiv.org/abs/2404.11455">Solution to the iterative differential equation -gamma*g' = g^{-1}</a>, arXiv:2404.11455 [math.CA], 2024.

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Roland Miyamoto and J. W. Sander, <a href="https://doi.org/10.1007/978-3-031-31617-3_14">Solving the iterative differential equation -gamma*g' = g^{-1}</a>, in: H. Maier, J. & R. Steuding (eds.), <a href="https://doi.org/10.1007/978-3-031-31617-3">Number Theory in Memory of Eduard Wirsing</a>, Springer, 2023, pp. 223-236, <a href="https://link.springer.com/chapter/10.1007/978-3-031-31617-3_14">alternative link</a>.

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Set kappa_n := A369990(n) / A369991(n) and theta_n := (kappa_n-kappa_{n+1}) / (kappa_{n-1}-kappa_n). Under the hypothesis that theta_{2m} < theta_{2m+2} < theta_{2*m+3} < theta_{2*m+1} for m=1,2,... (verified for all values known so far), we would obtain 0.27887706136895087 < kappa_{21}' < kappa < kappa_{22}' < 0.27887706136898083, which is sharper than formula (3) below. Here, the transformed sequence (kappa_n') = G(kappa_n) is defined via kappa_n' := (kappa_{n-1}*kappa_{n+1} - kappa_n^2) / (kappa_{n-1} - 2*kappa_n + kappa_{n+1}). (See the arXiv article for a proof of this conjecture-dependent statement.) Feeling even more adventurous, we can could apply the transformation G four times and would obtain 0.278877061368975064775 < kappa_{19}'''' < kappa < kappa_{18}'''' < 0.278877061368975064815.

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Set kappa_n := A369990(n) / A369991(n) and theta_n := (kappa_n-kappa_{n+1}) / (kappa_{n-1}-kappa_n). Under the hypothesis that theta_{2m} < theta_{2m+2} < theta_{2*m+3} < theta_{2*m+1} for m=1,2,... (verified for all values known so far), we would obtain 0.27887706136895087 < kappa_{21}' < kappa < kappa_{22}' < 0.27887706136898083, which is sharper than formula (3) below. Here, the transformed sequence (kappa_n') = G(kappa_n) is defined via kappa_n' := (kappa_{n-1}*kappa_{n+1} - kappa_n^2) / (kappa_{n-1} - 2*kappa_n + kappa_{n+1}). (See the arXiv article for a proof of this conjecture-dependent statement.) Feeling even more adventurous, we can apply the transformation G four times and obtain 0.278877061368975064775 < kappa_{19}'''' < kappa < kappa_{18}'''' < 0.278877061368975064815.

(See the arXiv article for a proof of this conjecture-dependent statement.) Feeling even more adventurous, we can apply the transformation G four times and obtain 0.278877061368975064775 < kappa_{19}'''' < kappa < kappa_{18}'''' < 0.278877061368975064815.