A332095 Numbers m such that 0 <= m*tan(m) < 1, ordered by |m|.

0, -3, 22, 44, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 312689, 1146408, 5419351, 10838702, -6167950454, -21053343141, -42106686282, -63160029423, -84213372564, -105266715705, -8958937768937, -17917875537874, -428224593349304, -856449186698608, -6134899525417045

Offset: 1

Author

M. F. Hasler, Sep 10 2020

Keywords

• sign

Comments

Equivalently, 0 together with integers m such that |tan(m)| < 1/m, multiplied by sign(tan(m)).

The term a(2) = 3 is up to 10^7 the only term m for which tan(m) < 0.

A092328 appears to be a subsequence. Does it contain all terms with tan(m) > 0?

Many terms are multiples of a smaller term: 44 = 22*2 and a(4..12) = {355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195} = 355*{1, 2, 3, ..., 9}.

Indeed, if |m*tan(m)| < 1/k^2 for some k = 1, 2, 3..., then (k*m)*tan(k*m) ~ k^2*m*tan(m) < 1. (E.g., for m = 355, m*tan(m) ~ 0.01.)

The "seeds" for which |m*tan(m)| 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), ...

Other terms in the sequence include: -21053343141*{1, 2, 3, 4, 5}, -8958937768937*{1, 2}, -6134899525417045, -66627445592888887, 430010946591069243, -2646693125139304345*{1, 2, 3, 4, 5}, ...

The absolute values of nonzero terms are a subsequence of A337371. - R. J. Mathar, Sep 24 2020

Can someone find a counterexample for which |sin(m)| < 1/m and |m*tan(m)| > 1? - M. F. Hasler, Oct 09 2020

Crossrefs

Cf. A092328, A088306, A337371 (similar, with sin, a superset except for the initial term).

Programs

PARI

is_A332095(n)={tan(n)*n < 1 && n*tan(n) >= 0}
for(n=0,oo, n*abs(tan(n))<1 && print1(sign(tan(n))*n", "))
/* Much faster: apply to numerators of convergents of Pi the function check(n) which prints all nonzero k*n in the sequence and returns the largest such k, largest in magnitude, possibly negative. N.B.: stops when (k+1)n is not in the sequence, so e.g., n = 11 (in convergents of Pi/2) does not give 22 and 44! */
print1(0); apply( {check(n)=for(i=1,oo,abs(i*n*tan(i*n))<1||return(sign(tan(n))*(i-1)); print1(", "sign(tan(n*i))*i*n))}, contfracpnqn(c=contfrac(Pi),#c)[1,]) \\ M. F. Hasler, Oct 09 2020