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.

13, 61, 169, 13321, 4621, 14197, 5041, 7345, 10261, 13861, 18217, 23401, 29485, 36541, 74401, 89761, 107101, 126541, 148201, 172201, 198661, 227701, 259441, 294001, 331501, 520885, 582121, 647977, 718621, 794221, 874945, 960961, 1052437, 1149541, 1252441, 1361305, 1898101, 2054053, 2218321

Offset: 2

Author

Robert Israel, Jan 05 2025

Keywords

• nonn
• base

Comments

a(n) == 1 mod n*(4*n^2-1) and is odd.

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000
• R. Israel and C. Lin, Sums of digits in three consecutive bases, Mathematics StackExchange, Jan 05 2025.

Crossrefs

Cf. A379749.

Programs

Maple

f:= proc(b) local a,x;
    for a from 1 by 2 to 10^7 do
      x:= a*(b^3-b)/2+1;
      if andmap(t -> convert(convert(x,base,t),`+`)=t, [b,b+1,b+2]) then return x fi
    od;
    FAIL
end proc:
map(f, [seq(b,b=3..200,2)]);