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Numerators of best lower approximates h/k to sqrt(k); complement of A355169.
2

%I #5 Jul 10 2022 16:43:54

%S 1,5,8,11,27,36,52,58,64,70,76,89,96,103,110,125,140,148,156,164,181,

%T 198,207,216,225,234,272,322,343,364,419,430,453,476,488,500,512,524,

%U 536,548,573,598,702,729,756,811,868,882,911,955,970,985,1000,1015

%N Numerators of best lower approximates h/k to sqrt(k); complement of A355169.

%C See A355159.

%F a(n) = floor(m^(3/2)), where m = A355159(n).

%e The first five best lower approximates (to sqrt(1), sqrt(3), sqrt(4), sqrt(5), sqrt(9), respectively) are 1/1, 5/3, 8/4, 11/5, 27/9; these are A355168(n)/A355159(n), for n=1..5..

%t u = Select[Range[300], N[FractionalPart[#^(3/2)]] < 1/2 &] (* A355159 *)

%t v = Select[Range[300], N[FractionalPart[#^(3/2)]] > 1/2 &] (* A355160 *)

%t Floor[u^(3/2)] (* A355168: numerators for fractions h/k <= sqrt(k) *)

%t Floor[v^(3/2)] (* A355169: numerators for fractions h/k > sqrt(k) *)

%Y Cf. A355159, A355160, A355169, A355115.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jun 26 2022