A163491 A fractal sequence (if we delete the first occurrence of n we get the sequence itself).

1, 1, 1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 5, 2, 4, 6, 3, 1, 7, 5, 2, 8, 4, 6, 9, 3, 1, 10, 7, 5, 11, 2, 8, 12, 4, 6, 13, 9, 3, 14, 1, 10, 15, 7, 5, 16, 11, 2, 17, 8, 12, 18, 4, 6, 19, 13, 9, 20, 3, 14, 21, 1, 10, 22, 15, 7, 23, 5, 16, 24, 11, 2, 25, 17, 8, 26, 12

Offset: 1

Author

Benoit Cloitre, Jul 29 2009

Keywords

• nonn

Comments

Write the positive integers with two spaces between each integer: 1,,,2,,,3,,,4,,,5,,,6,..., and fill undefined places with the sequence itself. A003602 is obtained by starting from 1,,2,,3,,4,,5,_,6,....

From Peter Munn, Aug 02 2020: (Start)

a(n) - 1 is the row of A083044 in which n occurs.

The m-th occurrence of m is at position A083045(m-1).

(End)

Examples

			  1,_,_,2,_,_,3,_,_,4,... -->
  1,1,_,2,_,_,3,_,_,4,... -->
  1,1,1,2,_,_,3,_,_,4,... -->
  1,1,1,2,1,_,3,_,_,4,... -->
  1,1,1,2,1,2,3,_,_,4,... -->
  1,1,1,2,1,2,3,_,_,4,... -->
  1,1,1,2,1,2,3,1,_,4,... -->
  1,1,1,2,1,2,3,1,2,4,... -->
  ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003602, A007494, A061419, A083044, A083045.

Ordinal transform of A087088.

Programs

Mathematica

a[n_] := a[n] = If[Mod[n, 3] == 1, (n+2)/3, a[Floor[2n/3]]];
Array[a, 100] (* Jean-François Alcover, Jan 10 2022 *)

PARI

a(n) = n+=2; my(q,r); while([q,r]=divrem(n,3); r, n-=q); q; \\ Kevin Ryde, Jan 16 2021

Python

def a(n): return (n+2)//3 if n%3==1 else a(n*2//3)
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Jan 16 2021

Formula

a(3n-2) = n.

A083044(a(n) - 1, A087088(n) - 1) = n. - Peter Munn, Aug 02 2020

From Rémy Sigrist, Jan 15 2021: (Start)

a(n+ceiling(n/2)) = a(n).

a(n) = 1 iff n belongs to A061419.

(End)

a(n) = (n+2)/3 if n == 1 (mod 3), otherwise a(n) = a(floor(n*2/3)). - Michael S. Branicky and Kevin Ryde, Jan 16 2021

Extensions

Terms after a(70) corrected by Jon E. Schoenfield, Nov 26 2015