A126759 a(0) = 1; a(2n) = a(n); a(3n) = a(n); otherwise write n = 6i+j, where j = 1 or 5 and set a(n) = 2i+2 if j = 1, otherwise a(n) = 2i+3.

1, 2, 2, 2, 2, 3, 2, 4, 2, 2, 3, 5, 2, 6, 4, 3, 2, 7, 2, 8, 3, 4, 5, 9, 2, 10, 6, 2, 4, 11, 3, 12, 2, 5, 7, 13, 2, 14, 8, 6, 3, 15, 4, 16, 5, 3, 9, 17, 2, 18, 10, 7, 6, 19, 2, 20, 4, 8, 11, 21, 3, 22, 12, 4, 2, 23, 5, 24, 7, 9, 13, 25, 2, 26, 14, 10, 8, 27, 6, 28, 3, 2, 15, 29, 4, 30, 16, 11, 5, 31, 3

Offset: 0

Author

N. J. A. Sloane, based on email from Miles Okazaki (milesokazaki(AT)gmail.com), Feb 18 2007

Keywords

• nonn
• nice
• hear

Comments

Invented by Miles Okazaki, who said: I was trying to write a composition that has the same melody going at several different speeds. If this sequence is mapped onto musical notes and you play every other term, you get the original sequence at half speed. If you play every third term, you again get the same melody. And every 4th term, 6th term, 8th term, 12th term, etc. yields the same result. The pattern generates itself, adding two new increasing integers every six terms.

The formula in the definition encapsulates this verbal description. - N. J. A. Sloane

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Crossrefs

One more than A126760.

Cf. A007310.

Programs

Haskell

a126759 n = a126759_list !! n
a126759_list = 1 : f 1 where
   f n = (case mod n 6 of 1 -> 2 * div n 6 + 2
                          5 -> 2 * div n 6 + 3
                          3 -> a126759 $ div n 3
                          _ -> a126759 $ div n 2) : f (n + 1)
-- Reinhard Zumkeller, May 23 2013

Maple

a:=proc(n) option remember; local i,j;
if n = 0 then RETURN(1); fi;
if n mod 2 = 0 then RETURN(a(n/2)); fi;
if n mod 3 = 0 then RETURN(a(n/3)); fi;
j := n mod 6; i := (n-j)/6;
if j = 1 then RETURN(2*i+2) else RETURN(2*i+3); fi;
end;
[seq(a(n),n=0..100)];

Mathematica

a[n_] := a[n] = Module[{i, j}, If[n == 0, Return[1]]; If[Mod[n, 2] == 0, Return[a[n/2]]]; If[Mod[n, 3] == 0, Return[a[n/3]]]; j = Mod[n, 6]; i = (n-j)/6; If[j == 1, Return[2*i+2], Return[2*i+3]]]; Table[a[n], {n, 0, 90}] (* Jean-François Alcover, Feb 11 2014, after Maple *)

PARI

a(n) = if (n, if (!(n%2), a(n/2), if (!(n%3), a(n/3), my(k=n%6); if (k==1, 2*(n\6)+2, 2*(n\6)+3))), 1); \\ Michel Marcus, Aug 06 2022

Scheme

(definec (A126759 n) (cond ((zero? n) 1) ((even? n) (A126759 (/ n 2))) ((zero? (modulo n 3)) (A126759 (/ n 3))) ((= 1 (modulo n 6)) (+ 2 (/ (- n 1) 3))) (else (+ 1 (/ (+ n 1) 3)))))
;; Antti Karttunen, Jan 28 2015

Formula

For k > 1: a(A007310(k-1)) = k and a(m) < k for m < A007310(k-1). - Reinhard Zumkeller, Jun 16 2008

For n > 0: a(A007310(n)) = n and a(m) < n for m < A007310(n). - Reinhard Zumkeller, May 23 2013

a(0) = 1, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 2, a(6n-1) = 2n + 1. [Essentially same as the original description, except the last clause expressed slightly differently.] - Antti Karttunen, Jan 28 2015

Extensions

Typo in definition corrected by Reinhard Zumkeller, Jun 16 2008