A364863 - OEIS

%I #8 Aug 14 2023 08:31:56

%S 0,21

%N Number of iterations of x -> x + min { k in A036301 | k > x } until an element of A036301 is reached, or -1 if this never happens, starting with n.

%C The question whether the iteration always reaches an element of A036301 was raised on the SeqFan list in 2018, with "closest" instead of "next larger" (element of A036301). In that case one has 0 < n < 56 as a trivial counterexample. It is still open to our knowledge.

%C The first unknown term is currently a(2). Starting with x =2 we reach x = 336917039990529107004169 after 72 iterations.

%H Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2023/06/balancedunbalanced-numbers-and.html">Balanced/Unbalanced numbers (and iterations)</a>, personal blog "Cinquante signes" on blogspot,com, June 23, 2023

%H M. F. Hasler and N. J. A. Sloane, in reply to E. Angelini, <a href="http://list.seqfan.eu/pipermail/seqfan/2018-December/070820.html">Re: Balanced / Unbalanced numbers</a>, Dec. 10, 2018

%F a(n) = 0 iff A071650(n) = 0, i.e., for all n in A036301.

%e a(0) = 0 because n = 0 is an element of A036301 and therefore no iteration is required to reach such an element.

%e The smallest nonzero element of A036301 is 112. Therefore, all smaller positive numbers 0 < n < 112 go to n + 112 under the first iteration of

%e   f: x -> x + min { k in A036301 | k > x }.

%e Under iterations of f, 1 -> 113 -> 234 -> 548 -> 1109 -> 2229 -> 4460 -> 8931 -> 17865 -> 35872 -> 71875 -> 143891 -> 287898 -> 575804 -> 1151810 -> 2303826 -> 4607657 -> 9215347 -> 18430735 -> 36861480 -> 73723189 -> 147446477 which is the first element of A036301 to be reached, after a(1) = 21 iterations.

%o (PARI) a(n) = for(k=0,oo, A071650(n) || return(k); n+=next_A036301(n))

%Y Cf. A036301, A071650.

%K nonn,base

%O 0,2

%A _M. F. Hasler_, Aug 11 2023