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A379896 a(n) is the least number k whose digit sums are 2*n-1, 2*n and 2*n+1 in bases 2*n-1, 2*n and 2*n+1 respectively.

%I A379896 #13 Jan 06 2025 04:15:06
%S A379896 13,61,169,13321,4621,14197,5041,7345,10261,13861,18217,23401,29485,
%T A379896 36541,74401,89761,107101,126541,148201,172201,198661,227701,259441,
%U A379896 294001,331501,520885,582121,647977,718621,794221,874945,960961,1052437,1149541,1252441,1361305,1898101,2054053,2218321
%N A379896 a(n) is the least number k whose digit sums are 2*n-1, 2*n and 2*n+1 in bases 2*n-1, 2*n and 2*n+1 respectively.
%C A379896 a(n) == 1 mod n*(4*n^2-1) and is odd.
%C A379896 It is impossible for k to have digit sums b, b+1 and b+2 in bases b, b+1 and b+2 if b is even: see link.
%H A379896 Robert Israel, <a href="/A379896/b379896.txt">Table of n, a(n) for n = 2..10000</a>
%H A379896 R. Israel and C. Lin, <a href="https://math.stackexchange.com/questions/5019550/sums-of-digits-in-three-consecutive-bases">Sums of digits in three consecutive bases</a>, Mathematics StackExchange, Jan 05 2025.
%e A379896 a(3) = 169 because 169 = 331_7 = 251_8 = 207_9 has digit sums 7, 8, 9 in bases 7, 8, 9 respectively, and 169 is the least number that works.
%p A379896 f:= proc(b) local a,x;
%p A379896     for a from 1 by 2 to 10^7 do
%p A379896       x:= a*(b^3-b)/2+1;
%p A379896       if andmap(t -> convert(convert(x,base,t),`+`)=t, [b,b+1,b+2]) then return x fi
%p A379896     od;
%p A379896     FAIL
%p A379896 end proc:
%p A379896 map(f, [seq(b,b=3..200,2)]);
%Y A379896 Cf. A379749.
%K A379896 nonn,base
%O A379896 2,1
%A A379896 _Robert Israel_, Jan 05 2025