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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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 multiplicity(n,2) is even, a(n) = 2n; otherwise a(n)=n/2, where multiplicity(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-Fran├žois Alcover, Jun 10 2015, after Alois P. Heinz *)

PROG

(Other) /* 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)} [From Carl R. White, Aug 23 2010]

CROSSREFS

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

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

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Last modified May 25 19:28 EDT 2017. Contains 287059 sequences.