A073675 Rearrangement of natural numbers such that a(n) is the smallest proper divisor of n not included earlier but if no such divisor exists then a(n) is the smallest proper multiple of n not included earlier, subject always to the condition that a(n) is not equal to n.

2, 1, 6, 8, 10, 3, 14, 4, 18, 5, 22, 24, 26, 7, 30, 32, 34, 9, 38, 40, 42, 11, 46, 12, 50, 13, 54, 56, 58, 15, 62, 16, 66, 17, 70, 72, 74, 19, 78, 20, 82, 21, 86, 88, 90, 23, 94, 96, 98, 25, 102, 104, 106, 27, 110, 28, 114, 29, 118, 120, 122, 31, 126, 128, 130, 33, 134, 136

Offset

1,1

Comments

The parity of the sequence is E,D,E,E,E,D,E,E,E,D,E,E,E,D,E,E,E,D,E,E,E,D,..., that is, an D followed by three E's from the second term onwards.

Closely related to A035263: if A035263(n) = 1, a(n) = 2n; otherwise a(n)=n/2. - Franklin T. Adams-Watters, Feb 02 2006

This permutation is self-inverse. This is the case r=2 of sequences where a(n)=floor(n/r) if floor(n/r)>0 and not already in the sequence, a(n) = floor(n*r) otherwise. All such sequences (for r>=1) are permutations of the natural numbers. - Franklin T. Adams-Watters, Feb 06 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Index entries for sequences that are permutations of the natural numbers

Formula

If valuation(n,2) is even, a(n) = 2n; otherwise a(n)=n/2, where valuation(n,2) = A007814(n) is the exponent of the highest power of 2 dividing n. - Franklin T. Adams-Watters, Feb 06 2006, Jul 31 2009

a(k*2^m) = k*2^(m+(-1)^m), m >= 0, odd k >= 1. - Carl R. White, Aug 23 2010

Maple

a:= proc(n) local i, m; m:=n;
      for i from 0 while irem(m, 2, 'r')=0 do m:=r od;
      m*2^`if`(irem(i, 2)=1, i-1, i+1)
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 10 2014

Mathematica

a[n_] := Module[{i, m = n}, For[i = 0, {q, r} = QuotientRemainder[m, 2]; r == 0, i++, m = q]; m*2^If[Mod[i, 2] == 1, i-1, i+1]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 10 2015, after Alois P. Heinz *)

Prog

(GNU bc) scale=0; for(n=1; n<=100; n++){m=0; for(k=n; !k%2; m++)k/=2; k*2^(m+(-1)^m)} /* Carl R. White, Aug 23 2010 */
(PARI) a(n) = if (valuation(n, 2) % 2, n/2, 2*n); \\ Michel Marcus, Mar 17 2018

Crossrefs

Matches A118967 for all non-powers-of-two. - Carl R. White, Aug 23 2010

Row 2 and column 2 of A059897.

Sequence in context: A011419 A011133 A197806 * A188167 A156034 A160581

Adjacent sequences: A073672 A073673 A073674 * A073676 A073677 A073678

Keyword

nonn

Author

Amarnath Murthy, Aug 11 2002

Extensions

More terms and comment from Franklin T. Adams-Watters, Feb 06 2006, Jul 31 2009

More terms from Franklin T. Adams-Watters, Feb 06 2006

Edited by N. J. A. Sloane, Jul 31 2009

Typo fixed by Charles R Greathouse IV, Apr 29 2010

Status

approved