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A189202 Let s_k(n) denote the sum of digits of n in base k. Then a(n) is the smallest m>0 such that both s_2(m*(n-1)) and s_n(2*m*(n-1))/(n-1) are even, or a(n)=0, if such m does not exist.

%I A189202 #25 May 20 2025 11:42:39
%S A189202 3,5,5,3,13,4,9,5,11,6,19,20,15,47,17,9,19,10,21,32,23,12,37,13,40,41,
%T A189202 29,15,46,16,33,17,35,18,37,56,39,20,41,21,85,22,45,68,47,72,73,25,51,
%U A189202 26,79,80,109,28,57,87,59,30,91,153,63,191,65,33,67,34,69
%N A189202 Let s_k(n) denote the sum of digits of n in base k. Then a(n) is the smallest m>0 such that both s_2(m*(n-1)) and s_n(2*m*(n-1))/(n-1) are even, or a(n)=0, if such m does not exist.
%C A189202 Conjecture: For all n>=2, a(n)>0.
%C A189202 For a general problem, see SeqFan link.
%H A189202 Alois P. Heinz, <a href="/A189202/b189202.txt">Table of n, a(n) for n = 2..10000</a>
%H A189202 Vladimir Shevelev, "<a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2011-April/014850.html">A new digital problem</a>", SeqFan Discussion, Apr 2011.
%p A189202 s:= proc(n, b) local m, t;
%p A189202       t:= 0; m:= n;
%p A189202       while m<>0 do t:= t+ irem(m, b, 'm') od; t
%p A189202     end:
%p A189202 a:= proc(n) local m;
%p A189202       for m while irem(s(m*(n-1), 2), 2)<>0 or
%p A189202                   irem(s(2*m*(n-1), n)/(n-1), 2)<>0 do od; m
%p A189202     end:
%p A189202 seq(a(n), n=2..100);  # _Alois P. Heinz_, May 02 2011
%t A189202 s[n_, b_] := Module[{m, t}, t = 0; m = n; While[m != 0 , t = t + Mod[m, b]; m = Quotient[m, b]]; t];
%t A189202 a[n_] := Module[{m}, For[m = 1, Mod[s[m*(n-1), 2], 2] != 0 || Mod[s[2*m*(n-1), n]/(n-1), 2] != 0, m++]; m];
%t A189202 Table[a[n], {n, 2, 100}](* _Jean-François Alcover_, May 20 2025, after _Alois P. Heinz_ *)
%Y A189202 Cf. A026741, A145051.
%K A189202 nonn
%O A189202 2,1
%A A189202 _Vladimir Shevelev_, May 02 2011