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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. 17
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
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
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 April 23 12:27 EDT 2024. Contains 371912 sequences. (Running on oeis4.)