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A332563 a(n) = minimal positive k such that the concatenation of the binary expansions of n,n+1,...,n+k is divisible by n+k+1, or -1 if no such k exists.

%I A332563 #44 Jul 09 2025 04:51:13
%S A332563 1,6,253,160,23,6,577,14,1,4,383,8,1591,18,169,42,1879,210,57,20,69,
%T A332563 1354,13,86,225,1532,577,300,13,30,6419,312,30639,12,151,8,89,2720,29,
%U A332563 5830,1,1450,195,478,55,166528,127,1074,3559,252,41
%N A332563 a(n) = minimal positive k such that the concatenation of the binary expansions of n,n+1,...,n+k is divisible by n+k+1, or -1 if no such k exists.
%C A332563 A base 2 analog of A332580.
%C A332563 For n up to 1000 the presently unknown values are a(213) and a(743).
%H A332563 Joseph Myers, <a href="/A332563/b332563.txt">Table of n, a(n) for n = 1..212</a>
%H A332563 Joseph Myers, <a href="/A332563/a332563_1.txt">Table of n, a(n) for n = 1..1024</a>. The UNKNOWN entries at n = 213 and 743 are either -1 or greater than 10^9. [This extends an earlier table of Scott R. Shannon, which searched up to 128 with a search limit of 10^6.]
%H A332563 J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, <a href="http://arxiv.org/abs/2004.14000">Three Cousins of Recaman's Sequence</a>, arXiv:2004:14000 [math.NT], April 2020.
%H A332563 Scott R. Shannon, <a href="/A332586/a332586_1.txt">The quotient after the final division, for n = 1..15</a>
%H A332563 N. J. A. Sloane, <a href="https://vimeo.com/457349959">Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows</a>, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk).
%Y A332563 Cf. A332580, A332584, A332586.
%K A332563 sign,base
%O A332563 1,2
%A A332563 _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 25 2020