cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A355115 a(n) = index of n^2 in A355159.

Original entry on oeis.org

1, 3, 5, 9, 16, 24, 29, 37, 44, 53, 66, 80, 94, 108, 121, 135, 155, 178, 197, 217, 235, 256, 285, 314, 337, 364, 388, 416, 452, 488, 522, 554, 584, 619, 660, 704, 743, 780, 815, 854, 901, 946, 990, 1036, 1077, 1120, 1174, 1229, 1283, 1336, 1384, 1434, 1494
Offset: 1

Views

Author

Clark Kimberling, Jun 26 2022

Keywords

Crossrefs

Cf. A355159.

Programs

  • Mathematica
    u = Select[Range[5000], N[FractionalPart[#^(3/2)]] < 1/2 &];  (* A355159 *)
Flatten[Position[FractionalPart[Sqrt[u]], 0]]  (* A355115 *)

A355160 Numbers k such that (fractional part of k^(3/2)) > 1/2.

Original entry on oeis.org

2, 6, 7, 8, 10, 12, 13, 19, 24, 26, 31, 33, 39, 40, 41, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 58, 60, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 89, 90, 93, 95, 96, 104, 105, 107, 109, 110, 117, 118, 120, 122, 124, 125, 132, 133, 135, 137
Offset: 1

Views

Author

Clark Kimberling, Jun 23 2022

Keywords

Comments

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

Crossrefs

Cf. A000093, A355159 (complement).

Programs

  • Mathematica
    Select[Range[300], N[FractionalPart[#^(3/2)]] < 1/2 &]  (* A355159 *)
Select[Range[300], N[FractionalPart[#^(3/2)]] > 1/2 &]  (* A355160 *)
  • PARI
    isok(k) = frac(k^(3/2)) > 1/2; \\ Michel Marcus, Jul 11 2022
  • Python
    from math import isqrt
from itertools import count, islice
def A355160_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)))
A355160_list = list(islice(A355160_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

Views

Author

Clark Kimberling, Jun 26 2022

Keywords

Comments

See A355159.

Examples

			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..

Crossrefs

Cf. A355159, A355160, A355169, A355115.

Programs

  • Mathematica
    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) *)

Formula

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

Views

Author

Clark Kimberling, Jun 26 2022

Keywords

Comments

See A355160.

Examples

			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.

Crossrefs

Cf. A355159, A355160, A355168, A355115.

Programs

  • Mathematica
    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) *)

Formula

a(n) = floor(m^(3/2)), where m = A355160(n).
Showing 1-4 of 4 results.

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