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A073675
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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.
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17
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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
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1,1
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COMMENTS
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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 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
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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