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A162319 Array read by antidiagonals: a(n,m) = the number of digits of m is when written in base n. The top row is the number of digits for each m in base 1.

%I A162319 #25 Aug 31 2017 21:59:24
%S A162319 1,1,2,1,2,3,1,1,2,4,1,1,2,3,5,1,1,1,2,3,6,1,1,1,2,2,3,7,1,1,1,1,2,2,
%T A162319 3,8,1,1,1,1,2,2,2,4,9,1,1,1,1,1,2,2,2,4,10,1,1,1,1,1,2,2,2,3,4,11,1,
%U A162319 1,1,1,1,1,2,2,2,3,4,12,1,1,1,1,1,1,2,2
%N A162319 Array read by antidiagonals: a(n,m) = the number of digits of m is when written in base n. The top row is the number of digits for each m in base 1.
%C A162319 A162320 is the array without the base 1 number lengths, and with the lengths of base 2 numbers in the top row.
%H A162319 Michael De Vlieger, <a href="/A162319/b162319.txt">Table of n, a(n) for n = 1..10440</a> (covers bases 1..144)
%e A162319 From _Michael De Vlieger_, Jan 02 2015: (Start)
%e A162319 Array read by antidiagonals begins:
%e A162319 1;
%e A162319 1, 2;
%e A162319 1, 2, 3;
%e A162319 1, 1, 2, 4;
%e A162319 1, 1, 2, 3, 5;
%e A162319 1, 1, 1, 2, 3, 6;
%e A162319 1, 1, 1, 2, 2, 3, 7;
%e A162319 1, 1, 1, 1, 2, 2, 3, 8;
%e A162319 1, 1, 1, 1, 2, 2, 2, 4, 9;
%e A162319 1, 1, 1, 1, 1, 2, 2, 2, 4, 10;
%e A162319 ...
%e A162319 Array adjusted such that the rows represent base n and the columns m:
%e A162319                        m
%e A162319          1  2  3  4  5  6  7  8  9  10
%e A162319          ------------------------------
%e A162319 base 1:  1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
%e A162319 base 2:  1, 2, 2, 3, 3, 3, 3, 4, 4, (4);
%e A162319 base 3:  1, 1, 2, 2, 2, 2, 2, 2, (3, 3);
%e A162319 base 4:  1, 1, 1, 2, 2, 2, 2, (2, 2, 2);
%e A162319 base 5:  1, 1, 1, 1, 2, 2, (2, 2, 2, 2);
%e A162319 base 6:  1, 1, 1, 1, 1, (2, 2, 2, 2, 2);
%e A162319 base 7:  1, 1, 1, 1, (1, 1, 2, 2, 2, 2);
%e A162319 base 8:  1, 1, 1, (1, 1, 1, 1, 2, 2, 2);
%e A162319 base 9:  1, 1, (1, 1, 1, 1, 1, 1, 2, 2);
%e A162319 base 10: 1, (1, 1, 1, 1, 1, 1, 1, 1, 1);
%e A162319 ...
%e A162319 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 = 4. The number m = 2 in base n = 4 requires 1 digit, thus a(12) = 1.
%e A162319 For n = 14, a(14) is found in the fourth position in row 5 in the array read by antidiagonals. This equates to m = 4, base n = 2. The number m = 4 in base n = 2 requires 3 digits, thus a(14) = 3. (End)
%t A162319 Table[Function[k, If[k == 1, m, IntegerLength[m, k]]][k - m + 1], {k, 13}, {m, k}] // Flatten (* _Michael De Vlieger_, Aug 31 2017 *)
%Y A162319 Cf. A162320.
%K A162319 base,nonn,tabl
%O A162319 1,3
%A A162319 _Leroy Quet_, Jul 01 2009