A046959 -id:A046959 - cp's OEIS frontend

A228784 Permutation of the positive integers related to the "Sine tree" (see Comments lines for construction details).

1, 3, 2, 4, 8, 9, 5, 7, 10, 18, 16, 37, 21, 15, 43, 75, 33, 36, 20, 14, 11, 19, 17, 34, 38, 22, 29, 41, 73, 66, 150, 86, 31, 172, 300, 132, 147, 83, 59, 44, 76, 68, 35, 39, 6, 13, 79, 71, 136, 152, 88, 119, 167, 295, 264, 600, 344, 62, 173, 301, 133, 146, 82

Offset: 1

Paul Tek, Oct 04 2013

The "Sine tree" is constructed according to this procedure:
   take an infinite complete binary tree,
   initially, the nodes have no value,
   for each n=1,2,3,...:
      move to the root node,
      while the current node has a value:
         if sin(n)         else move to the right child node.
      assign the value n to the current node.
As the set {sin(1), sin(2), sin(3), ...} is dense in the open interval ]-1, +1[, each node will eventually have a value.
a(n) corresponds to the "index" of the node with value n:
- the index of the root node is 1,
- the index of the left child of the node with index k is 2*k,
- the index of the right child of the node with index k is 2*k+1.
a(A046959(n)) = 2^(n-1)-1, for any n>1.
a(A046964(n)) = 2^(n-1), for any n>0.

			For n=1: the root node has no value, so we assign it the value 1, and a(1)=1.
For n=2: the root node has value 1, and sin(2)>sin(1), so we move to the right child node. This node has no value, so we assign it the value 2, and a(2)=2*1+1.
For n=3: the root node has value 1, and sin(3)<sin(1), so we move to the left child node. This node has no value, so we assign it the value 3, and a(3)=2*1.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, The 7 first nodes of the Sine tree
Paul Tek, Perl program for this sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A046959, A046964, A229976.

Perl
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See Links section.
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A229976 Inverse permutation to A228784.

Original entry on oeis.org

1, 3, 2, 4, 7, 45, 8, 5, 6, 9, 21, 310, 46, 20, 14, 11, 23, 10, 22, 19, 13, 26, 334, 378, 104038, 89, 309, 335, 27, 341, 33, 344, 17, 24, 43, 18, 12, 25, 44, 336

Offset: 1

Paul Tek, Oct 05 2013

nonn

a(A228784(n))=n, for any n>0.
a(2^n-1)=A046959(n+1), for any n>0.
a(2^n)=A046964(n+1), for any n>=0.

Cf. A046959, A046964, A228784.

A172451 a(1) = 1, and for each n >= 2, a(n) is the smallest number such that 1/sin(a(n)) < 1/sin(a(k)) for all k < n, so that 1/sin(a(1)) > 1/sin(a(2)) > ... > 1/sin(a(n)) > ...

Original entry on oeis.org

1, 2, 4, 6, 22, 333, 355, 103993, 104348, 1042060, 1146408, 4272943, 5419351, 80143857

Offset: 1

Michel Lagneau, Feb 03 2010

			1/sin(1) = 1.1883951; 1/sin(2) = 1.0997501; 1/sin(4) = - 1.3213487.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.

Eric Weisstein's World of Mathematics, Pi

Cf. A002485, A046959, A046965, A046964, A172445, A172446.

a:= evalf(1/sin(1)); for n from 2 to 10000000 do; if a > evalf(1/sin(n)) then a:= evalf(1/sin(n)); print(n); else fi ; od;

vm = 2; s = {}; Do[v = 1/Sin[n]; If[v < vm, vm = v; AppendTo[s, n]], {n, 1, 110000}]; s (* Amiram Eldar, Aug 10 2019 *)

lista(NN) = {my(x=2); for(k=1, NN, if(1/sin(k)Jinyuan Wang, Aug 12 2019

a(13) corrected and a(14) added by Amiram Eldar, Aug 10 2019

A308879 a(n) is the n-digit integer m that maximizes sin(m).

Original entry on oeis.org

8, 33, 699, 9929, 51819, 573204, 4846147, 37362253, 288632526, 9251925681, 81129397337, 881156436695

Offset: 1

Inigo Quilez, Feb 12 2020

a(n) is also the n-digit integer that minimizes the mean square error of the approximation sin(x+m) for cos(x) over [0, 2*Pi].

Naturally, sin(a(n)) is the best approximation to 1 for an n-digit integer argument. a(n) is the closest integer to an n-digit number of the form (4k+1)*Pi/2. Often used to compute an approximated rotation matrix with just a few number of characters of code, as in M = sin(x+{0,699,-699,0}). It is not guaranteed that each term in the sequence produces a better approximation than the previous one, although numerical evidence suggests so. It is therefore also not guaranteed to be a subsequence of A046959.

			For n=3, a(3)=699 since no other 3-digit integer m makes sin(x+m) closer to cos(x) than m=699 does. For example, cos(4.5) = -0.210795799... and sin(4.5+699) = -0.215061112... and no other value of m will make the latter closer to the former.

Cf. A046959.

double e = 1.0;
  int b = 0, d=1, c=10;
  int a[10]; // print A to see the results
  for( int i=0; d<10; i++ )
  {
      double y = double(i*4+1)*PI/2.0;
      double z = round(y);
      double f = abs(z-y);
      int    w = int(z);
      if( w>=c ) { a[d]=b; c*=10; e=1.0; b=0; d++; }
      if( f< e ) { e=f; b=w; }
  }

A385636 Numbers k where sin(sqrt(k)) reaches successive record high values.

Original entry on oeis.org

0, 1, 2, 60, 61, 62, 200, 417, 2687, 1536009, 57157270, 81436635, 143005063, 706132910, 2016118929

Offset: 1

Artur Jasinski, Jul 05 2025

Each record value is ever closer to 1, but never reaches 1 since Pi is not algebraic.

			         n sin(sqrt(n))
         0 0.00000000000000000000000000000000000000
         1 0.84147098480789650665250232163029899962
         2 0.98776594599273552706913407207894265590
        60 0.99417205583623141588832072787265937649
        61 0.99904391031188045397420593007730566087
        62 0.99979948155661636906716009313812999866
       200 0.99998765612212040379361805662023274661
       417 0.99999997455044116794629686048124083679
      2687 0.99999999999810547903298251324014875470
   1536009 0.99999999999999044600166587966774547198
  57157270 0.99999999999999422252795052032516595669
  81436635 0.99999999999999964054666338696864815400
 143005063 0.99999999999999999572901646060974247156
 706132910 0.99999999999999999879205185535546721950
2016118929 0.99999999999999999996673765988843841836

Cf. A046959, A046964, A046965.

max = -2; aa = {}; Do[kk = N[Sin[Sqrt[n]], 50]; If[kk > max, max = kk; Print[{n, kk}]; AppendTo[aa, n]], {n, 0, 150000000}]; aa

a(14)-a(15) from David Consiglio, Jr., Jul 14 2025

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A046964 Sin(n) decreases monotonically to -1.

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A228784 Permutation of the positive integers related to the "Sine tree" (see Comments lines for construction details).

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A229976 Inverse permutation to A228784.

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A172451 a(1) = 1, and for each n >= 2, a(n) is the smallest number such that 1/sin(a(n)) < 1/sin(a(k)) for all k < n, so that 1/sin(a(1)) > 1/sin(a(2)) > ... > 1/sin(a(n)) > ...

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A308879 a(n) is the n-digit integer m that maximizes sin(m).

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C

A385636 Numbers k where sin(sqrt(k)) reaches successive record high values.

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Mathematica

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