A136498 -id:A136498 - cp's OEIS frontend

A138330 Beatty discrepancy (defined in A138253) giving the closeness of the pair (A136497,A136498) to the Beatty pair (A001951,A001952).

1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Mar 14 2008

nonn

Old definition was "Beatty discrepancy of the complementary equation b(n) = a(a(n)) + a(n)".

			d(1) - c(c(1)) - c(1) =  3 - 1 - 1 = 1;
d(2) - c(c(2)) - c(2) =  6 - 2 - 2 = 2;
d(3) - c(c(3)) - c(3) = 10 - 5 - 4 = 1;
d(4) - c(c(4)) - c(4) = 13 - 7 - 5 = 1.

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Cf. A001951, A001952, A136497, A136498, A138253, A059648.

[2*n - Floor(Sqrt(2)*Floor(Sqrt(2)*n)): n in [1..100]]; // Vincenzo Librandi, Nov 12 2018

a:=n->2*n-floor(sqrt(2)*floor(sqrt(2)*n)): seq(a(n),n=1..120); # Muniru A Asiru, Nov 11 2018

Table[2 n - Floor[Sqrt[2] Floor[Sqrt[2] n]], {n, 1, 100}] (* Vincenzo Librandi, Nov 12 2018 *)

a(n)=2*n-floor(sqrt(2)*floor(sqrt(2)*n)) \\ Benoit Cloitre, May 08 2008

from math import isqrt
def A138330(n): return (m:=n<<1)-isqrt(isqrt(n*m)**2<<1) # Chai Wah Wu, Aug 29 2022

a(n) = d(n) - c(c(n)) - c(n), where c(n) = A001951 and d(n) = A001952.
a(n) = 2*n - A007069(n). - Benoit Cloitre, May 08 2008
a(n) = A059648(n+1) + 1. - Michel Dekking, Nov 11 2018

Definition revised by N. J. A. Sloane, Dec 16 2018

A136497 Solution of the complementary equation b(n)=a(a(n))+a(n).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 10, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 30, 32, 33, 34, 35, 37, 39, 41, 43, 44, 46, 47, 48, 50, 51, 53, 54, 56, 58, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 79, 81, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 96, 97, 98, 99

Offset: 1

Clark Kimberling, Jan 01 2008

nonn

			b(1) = a(a(1))+a(1) = 1+1 = 2;
b(2) = a(a(2))+a(2) = 4+3 = 7;
b(3) = a(a(3))+a(3) = 5+4 = 9.

Clark Kimberling, "Complementary Equations," Journal of Integer Sequences 10 (2007) Article 07.1.4, 1-14.

Cf. A136498.

A384662 Solution of the complementary equation b(n)=a(a(n))+a(n)+2 with a(1)=1; this is the sequence b.

Original entry on oeis.org

4, 6, 8, 14, 19, 23, 25, 28, 30, 32, 37, 39, 41, 44, 49, 52, 55, 60, 64, 67, 73, 78, 82, 84, 87, 89, 94, 99, 103, 106, 110, 113, 115, 118, 122, 124, 129, 131, 135, 138, 140, 142, 148, 150, 153, 158, 160, 165, 167, 169, 171, 174, 178, 181, 183, 186, 190, 193

Offset: 1

Clark Kimberling, Jun 09 2025

nonn

Sequence a is A384661.

			b(1) = a(a(1))+a(1)+2 = 1+1+2 = 4;
b(2) = a(a(2))+a(2)+2 = 2+2+2 = 6;
b(3) = a(a(3))+a(3)+2 = 3+3+2 = 8;
b(4) = a(a(4))+a(4)+2 = 5+7+2 = 14.

Clark Kimberling, Complementary equations, J. Int. Seq. Article 07.1.4 (2007), 1-13.

Cf. A136498, A136499, A136500, A384661, A384663, A384664.

{b(n)-b(n-1) : n>=2} = {2, 3, 4, 5, 6}.

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A138330 Beatty discrepancy (defined in A138253) giving the closeness of the pair (A136497,A136498) to the Beatty pair (A001951,A001952).

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A136497 Solution of the complementary equation b(n)=a(a(n))+a(n).

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A384662 Solution of the complementary equation b(n)=a(a(n))+a(n)+2 with a(1)=1; this is the sequence b.

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