Revision History for A332095
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Can someone find a counter-example counterexample for which |sin(m)| < 1/m and |m*tan(m)| > 1? - M. F. Hasler, Oct 09 2020
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Numbers n m such that 0 <= nm*tan(nm) < 1, ordered by |nm|.
Equivalently, 0 together with integers m such that |tan(nm)| < 1/n, m, multiplied by sign(tan(nm)).
The term a(2) = 3 is up to 10^7 the only term n m for which tan(nm) < 0.
A092328 appears to be a subsequence. Does it contain all terms with tan(nm) > 0 ?
Indeed, if |nm*tan(nm)| < 1/k^2 for some k = 1, 2, 3..., then (k*nm)*tan(k*nm) ~ k^2*nm*tan(nm) < 1. (E.g., for n m = 355, nm*tan(nm) ~ 0.01.)
The "seeds" for which |nm*tan(nm)| is particularly small are numerators of convergents of continued fractions for Pi (A002485) (and/or Pi/2: A096456), e.g., a(3) = numerator(22/7), a(5) = numerator(355/113), ...
Can someone find a counter-example for which |sin(nm)| < 1/n m and |nm*tan(nm)| > 1? - M. F. Hasler, Oct 09 2020
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Equivalently, 0 and together with integers such that |tan(n)| < 1/n, multiplied by sign(tan(n)).
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