cp's OEIS Frontend

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.

%I A073675 #45 Aug 08 2025 12:30:37
%S A073675 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,
%T A073675 54,56,58,15,62,16,66,17,70,72,74,19,78,20,82,21,86,88,90,23,94,96,98,
%U A073675 25,102,104,106,27,110,28,114,29,118,120,122,31,126,128,130,33,134,136
%N 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.
%C A073675 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.
%C A073675 Closely related to A035263: if A035263(n) = 1, a(n) = 2n; otherwise a(n)=n/2. - _Franklin T. Adams-Watters_, Feb 02 2006
%C A073675 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
%C A073675 Take the list of positive integers L. At each step n swap L(n) with L(2*L(n)). - _Ali Sada_, Jun 18 2025
%H A073675 Alois P. Heinz, <a href="/A073675/b073675.txt">Table of n, a(n) for n = 1..1000</a>
%H A073675 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A073675 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
%F A073675 a(k*2^m) = k*2^(m+(-1)^m), m >= 0, odd k >= 1. - _Carl R. White_, Aug 23 2010
%p A073675 a:= proc(n) local i, m; m:=n;
%p A073675       for i from 0 while irem(m, 2, 'r')=0 do m:=r od;
%p A073675       m*2^`if`(irem(i, 2)=1, i-1, i+1)
%p A073675     end:
%p A073675 seq(a(n), n=1..80);  # _Alois P. Heinz_, Feb 10 2014
%t A073675 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_ *)
%o A073675 (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 */
%o A073675 (PARI) a(n) = if (valuation(n, 2) % 2, n/2, 2*n); \\ _Michel Marcus_, Mar 17 2018
%o A073675 (Python)
%o A073675 def A073675(n): return n>>1 if (~n & n-1).bit_length()&1 else n<<1 # _Chai Wah Wu_, Aug 08 2025
%Y A073675 Matches A118967 for all non-powers-of-two. - _Carl R. White_, Aug 23 2010
%Y A073675 Row 2 and column 2 of A059897.
%K A073675 nonn
%O A073675 1,1
%A A073675 _Amarnath Murthy_, Aug 11 2002
%E A073675 More terms and comment from _Franklin T. Adams-Watters_, Feb 06 2006, Jul 31 2009
%E A073675 More terms from _Franklin T. Adams-Watters_, Feb 06 2006
%E A073675 Edited by _N. J. A. Sloane_, Jul 31 2009
%E A073675 Typo fixed by _Charles R Greathouse IV_, Apr 29 2010