

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
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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.
This permutation is selfinverse. 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. AdamsWatters, 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. AdamsWatters, 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, i1, i+1)
end:


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, i1, i+1]]; Table[a[n], {n, 1, 80}] (* JeanFrançois 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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



