cp's OEIS Frontend

A364863 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.

%I A364863 #13 Dec 23 2024 14:53:46
%S A364863 0,21
%N A364863 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 A364863 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 A364863 The first unknown term is currently a(2). Starting with x =2 we reach x = 336917039990529107004169 after 72 iterations.
%H A364863 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 A364863 Eric Angelini, <a href="/A364863/a364863.pdf">Balanced/Unbalanced numbers (and iterations)</a>, personal blog "Cinquante signes" on blogspot,com, June 23, 2023 [Cached copy]
%H A364863 M. F. Hasler and N. J. A. Sloane, in reply to E. Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/pipermail/seqfan/2018-December/070820.html">Re: Balanced / Unbalanced numbers</a>, Dec. 10, 2018
%F A364863 a(n) = 0 iff A071650(n) = 0, i.e., for all n in A036301.
%e A364863 a(0) = 0 because n = 0 is an element of A036301 and therefore no iteration is required to reach such an element.
%e A364863 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 A364863   f: x -> x + min { k in A036301 | k > x }.
%e A364863 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 A364863 (PARI) a(n) = for(k=0,oo, A071650(n) || return(k); n+=next_A036301(n))
%Y A364863 Cf. A036301, A071650.
%K A364863 nonn,base
%O A364863 0,2
%A A364863 _M. F. Hasler_, Aug 11 2023