A060833 Separate the natural numbers into disjoint sets A, B with 1 in A, such that the sum of any 2 distinct elements of the same set never equals 2^k + 2. Sequence gives elements of set A.

1, 4, 7, 8, 12, 13, 15, 16, 20, 23, 24, 25, 28, 29, 31, 32, 36, 39, 40, 44, 45, 47, 48, 49, 52, 55, 56, 57, 60, 61, 63, 64, 68, 71, 72, 76, 77, 79, 80, 84, 87, 88, 89, 92, 93, 95, 96, 97, 100, 103, 104, 108, 109, 111, 112, 113, 116, 119, 120, 121, 124, 125, 127, 128

Offset: 1

Author

Sen-Peng Eu, May 01 2001

Keywords

• easy
• nonn

Comments

Can be constructed as follows: place of terms of (2^k+1,2^k+2,...,2^k) are the reflection from (2,3,4,...,2^k,1). [Comment not clear to me - N. J. A. Sloane]

If n == 0 mod 4, then n is in the sequence. If n == 2 mod 4, then n is not in the sequence. The number 2n - 1 is in the sequence if and only if n is in the sequence. For n > 1, n is in the sequence if and only if A038189(n-1) = 1. - N. Sato, Feb 12 2013

The set B contains all numbers 2^(k-1)+1 = (2^k+2)/2 (half of the "forbidden sums"), (2, 3, 5, 9, 17, 33, 65,...) = 1/2 * (4, 6, 10, 18, 34, 66, 130, 258,...). - M. F. Hasler, Feb 12 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Kevin Ryde, Iterations of the Dragon Curve, see index TurnRight, with a(n) = TurnRight(n-2) + 1 for n>=2.

Crossrefs

Essentially one more than A091067.

First differences: A106836.

A082410(a(n)) = 0.

Programs

Maple

a:= proc(n) option remember; local k, t;
      if n=1 then 1
    else for k from 1+a(n-1) do t:= k-1;
           while irem(t, 2, 'r')=0 do t:=r od;
           if irem(t, 4)=3 then return k fi
         od
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 12 2013

Mathematica

a[n_] := a[n] = Module[{k, t, q, r}, If[n == 1, 1, For[k = 1+a[n-1], True, k++, t = k-1; While[{q, r} = QuotientRemainder[t, 2]; r == 0, t = q]; If[Mod[t, 4] == 3, Return[k]]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 30 2017, after Alois P. Heinz *)

PARI

a(n) = if(n=2*n-2, my(t=1); forstep(i=logint(n,2),0,-1, if(bittest(n,i)==t, n++;t=!t))); n+1; \\ Kevin Ryde, Mar 21 2021

Formula

a(1) = 1; and for n > 1: a(n) = A091067(n-1)+1. - Antti Karttunen, Feb 20 2015, based on N. Sato's Feb 12 2013 comment above.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 10 2001