A109852 a(1)=1; a(2) = 2; for n >= 1 and 1 <= k < 2^n, a(2^n+k) is the least multiple of a(2^n-k) not included earlier and a(2^n) is the least number not included earlier.

1, 2, 3, 4, 6, 8, 5, 7, 10, 16, 12, 20, 9, 14, 11, 13, 22, 28, 18, 40, 24, 32, 30, 21, 15, 48, 36, 44, 27, 26, 17, 19, 34, 52, 54, 88, 72, 96, 45, 42, 60, 64, 120, 80, 90, 56, 66, 39, 33, 70, 63, 100, 84, 112, 50, 35, 25, 104, 78, 68, 51, 38, 23, 29, 46, 76, 102, 136, 156, 208

Offset: 1

Author

Amarnath Murthy, Jul 07 2005

Keywords

• nonn
• look

Comments

Every number appears and no number is repeated.

Conjecture: a(2^n) is prime if n is not 0 nor 2.

Conjecture: for n>2, every odd prime >4 is encountered in order at a(2^n-1), a(2^n). - Bill McEachen, May 06 2014

Examples

			a(8) = 7 as the least number not included earlier. a(9) = 2*a(8) = 2*5=10,
a(10) = 2*a(6) = 16, a(11) = 2*a(5) = 12, a(12)= 5*a(4) = 20 as 8, 12 and 16 have already been included.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A109853, A308301 (inverse).

Programs

Maple

did := [1]; lef := []; for n from 2 to 1000 do lef := [op(lef),n]; od : tak2n := proc(n2n) local i; global lef; i := op(1,lef); lef := subsop(1=NULL,lef); RETURN(i); end : tak := proc(n2n) local noffs,need,lefi,nindx,aa,mul; global lef,did; for noffs from -1 to -n2n+1 by -1 do nindx := n2n+noffs; aa := did[nindx]; for mul from 2 to 10000 do need := aa*mul; if member(need,lef,'lefi') = true then break; fi; od : lef := subsop(lefi=NULL,lef); printf("%d,",need); did := [op(did),need]; od : RETURN(ret); end : printf("1,"); for bas from 1 to 5 do nstrt := 2^bas; a := tak2n(nstrt); printf("%d,",a); did := [op(did),a]; tak(nstrt); od : # R. J. Mathar, Mar 27 2006
# second Maple program:
ina:= proc(n) evalb(n<3) end:
a:= proc(n) option remember; local k, i, t;
      if n<3 then n
      else a(n-1);
           k:= n-2^ilog2(n);
           t:= `if`(k=0, 1, a(n-2*k));
           for i from 2*t by t while ina(i) do od;
           ina(i):= true; i
      fi
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 07 2011

Mathematica

f[s_] := Block[{k = 2, len = Length@s}, exp = Ceiling[Log[2, len]]; m = s[[2^exp - len + 1]]; While[MemberQ[s, k*m], k++ ]; Append[s, k*m]]; Rest@Nest[f, {1, 1}, 70] (* the programming trick is to set a(0)=1 *) (* Robert G. Wilson v *)

Extensions

More terms from R. J. Mathar, Mar 27 2006

Edited and extended by Robert G. Wilson v, Jun 14 2006