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A162320 Array read by antidiagonals: a(n,m) = the number of digits of m when written in base n. The top row is the number of digits for each m in base 2.

%I A162320 #25 Jun 10 2025 07:57:40
%S A162320 1,1,2,1,1,2,1,1,2,3,1,1,1,2,3,1,1,1,2,2,3,1,1,1,1,2,2,3,1,1,1,1,2,2,
%T A162320 2,4,1,1,1,1,1,2,2,2,4,1,1,1,1,1,2,2,2,3,4,1,1,1,1,1,1,2,2,2,3,4,1,1,
%U A162320 1,1,1,1,2,2,2,2,3,4,1,1,1,1,1,1,1,2,2
%N A162320 Array read by antidiagonals: a(n,m) = the number of digits of m when written in base n. The top row is the number of digits for each m in base 2.
%C A162320 A162319 is the same array with the lengths of base 1 numbers in the top row.
%H A162320 Michael De Vlieger, <a href="/A162320/b162320.txt">Table of n, a(n) for n = 1..10296</a> (Covers bases n = 1..144)
%e A162320 From _Michael De Vlieger_, Jan 02 2015: (Start)
%e A162320 Array read by antidiagonals begins:
%e A162320   1;
%e A162320   1, 2;
%e A162320   1, 1, 2;
%e A162320   1, 1, 2, 3;
%e A162320   1, 1, 1, 2, 3;
%e A162320   1, 1, 1, 2, 2, 3;
%e A162320   1, 1, 1, 1, 2, 2, 3;
%e A162320   1, 1, 1, 1, 2, 2, 2, 4;
%e A162320   1, 1, 1, 1, 1, 2, 2, 2, 4;
%e A162320   ...
%e A162320 Array adjusted such that the rows represent base n and the columns m:
%e A162320                          m
%e A162320            1  2  3  4  5  6  7  8  9  10
%e A162320            ------------------------------
%e A162320   base 2:  1, 2, 2, 3, 3, 3, 3, 4, 4, (4);
%e A162320   base 3:  1, 1, 2, 2, 2, 2, 2, 2, (3, 3);
%e A162320   base 4:  1, 1, 1, 2, 2, 2, 2, (2, 2, 2);
%e A162320   base 5:  1, 1, 1, 1, 2, 2, (2, 2, 2, 2);
%e A162320   base 6:  1, 1, 1, 1, 1, (2, 2, 2, 2, 2);
%e A162320   base 7:  1, 1, 1, 1, (1, 1, 2, 2, 2, 2);
%e A162320   base 8:  1, 1, 1, (1, 1, 1, 1, 2, 2, 2);
%e A162320   base 9:  1, 1, (1, 1, 1, 1, 1, 1, 2, 2);
%e A162320   base 10: 1, (1, 1, 1, 1, 1, 1, 1, 1, 1);
%e A162320   ...
%e A162320 For n = 12, a(12) is found in the second position in row 5 in the array read by antidiagonals. This equates to m = 2, base n = 5. The number m = 2 in base n = 5 requires 1 digit, thus a(12) = 1.
%e A162320 For n = 20, a(20) is found in the fifth position in row 6 in the array read by antidiagonals. This equates to m = 5, base n = 3. The number m = 5 in base n = 3 requires 2 digits, thus a(20) = 2. (End)
%t A162320 a162320[n_] := Block[{t = {}, i, j}, For[i = 1, i <= n, i++, For[j = i, j > 1, j--, AppendTo[t, Floor@Log[j, i - j + 1] + 1]]]; t]; a162320[14] (* _Michael De Vlieger_, Jan 02 2015 *)
%Y A162320 Cf. A162319.
%K A162320 base,nonn,tabl
%O A162320 1,3
%A A162320 _Leroy Quet_, Jul 01 2009