A355160 -id:A355160 - cp's OEIS frontend

A355159 Numbers k such that (fractional part of k^(3/2)) < 1/2.

0, 1, 3, 4, 5, 9, 11, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 27, 28, 29, 30, 32, 34, 35, 36, 37, 38, 42, 47, 49, 51, 56, 57, 59, 61, 62, 63, 64, 65, 66, 67, 69, 71, 79, 81, 83, 87, 91, 92, 94, 97, 98, 99, 100, 101, 102, 103, 106, 108, 111, 112, 113, 114, 115

Offset: 0

Clark Kimberling, Jun 22 2022

For each nonnegative integer K there is a greatest nonnegative integer h such that h/K <= sqrt(K); a(n) is the n-th number h such that h/K is closer to sqrt(K) than (h+1)/K is.

Cf. A000093, A355160 (complement).

Select[-1 + Range[300], N[FractionalPart[#^(3/2)]] < 1/2 &]  (* A355159 *)
Select[-1 + Range[300], N[FractionalPart[#^(3/2)]] > 1/2 &]  (* A355160 *)

isok(k) = frac(k^(3/2)) < 1/2; \\ Michel Marcus, Jul 11 2022

from math import isqrt
from itertools import count, islice
def A355159_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:int(((r:=n**3)-(m:=isqrt(r))*(m+1))<<2<=1),count(max(startvalue,0)))
A355159_list = list(islice(A355159_gen(),30)) # Chai Wah Wu, Aug 03 2022

A355168 Numerators of best lower approximates h/k to sqrt(k); complement of A355169.

Original entry on oeis.org

1, 5, 8, 11, 27, 36, 52, 58, 64, 70, 76, 89, 96, 103, 110, 125, 140, 148, 156, 164, 181, 198, 207, 216, 225, 234, 272, 322, 343, 364, 419, 430, 453, 476, 488, 500, 512, 524, 536, 548, 573, 598, 702, 729, 756, 811, 868, 882, 911, 955, 970, 985, 1000, 1015

Offset: 1

Clark Kimberling, Jun 26 2022

See A355159.

			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..

Cf. A355159, A355160, A355169, A355115.

u = Select[Range[300], N[FractionalPart[#^(3/2)]] < 1/2 &]  (* A355159 *)
v = Select[Range[300], N[FractionalPart[#^(3/2)]] > 1/2 &]  (* A355160 *)
Floor[u^(3/2)]  (* A355168: numerators for fractions h/k <= sqrt(k) *)
Floor[v^(3/2)]  (* A355169: numerators for fractions h/k > sqrt(k) *)

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

A355169 Numbers h such that (h+1)/k is closer to sqrt(k) than h/k is, where h is the greatest integer j such that j/k < sqrt(k); complement of A355168.

Original entry on oeis.org

2, 14, 18, 22, 31, 41, 46, 82, 117, 132, 172, 189, 243, 252, 262, 281, 291, 301, 311, 332, 353, 374, 385, 396, 407, 441, 464, 560, 585, 610, 623, 636, 649, 662, 675, 688, 715, 742, 769, 783, 797, 825, 839, 853, 896, 925, 940, 1060, 1075, 1106, 1137, 1153

Offset: 1

Clark Kimberling, Jun 26 2022

See A355160.

			a(1) = 2 corresponds to 2/2 < sqrt(2) < 3/2.
a(2) = 14 corresponds to 14/6 < sqrt(6) < 15/6.
a(3) = 18 corresponds to 18/7 < sqrt(7) < 19/7.

Cf. A355159, A355160, A355168, A355115.

u = Select[Range[300], N[FractionalPart[#^(3/2)]] < 1/2 &]  (* A355159 *)
v = Select[Range[300], N[FractionalPart[#^(3/2)]] > 1/2 &]  (* A355160 *)
Floor[u^(3/2)]  (* A355168: numerators for fractions h/k <= sqrt(k) *)
Floor[v^(3/2)]  (* A355169: numerators for fractions h/k > sqrt(k) *)

a(n) = floor(m^(3/2)), where m = A355160(n).

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A355159 Numbers k such that (fractional part of k^(3/2)) < 1/2.

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A355168 Numerators of best lower approximates h/k to sqrt(k); complement of A355169.

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A355169 Numbers h such that (h+1)/k is closer to sqrt(k) than h/k is, where h is the greatest integer j such that j/k < sqrt(k); complement of A355168.

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