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.

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, 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, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 12, 1, 1, 1, 1, 1, 1, 2, 2

Offset

1,3

Comments

A162320 is the array without the base 1 number lengths, and with the lengths of base 2 numbers in the top row.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10440 (covers bases 1..144)

Example

From Michael De Vlieger, Jan 02 2015: (Start)
Array read by antidiagonals begins:
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, 3, 8;
1, 1, 1, 1, 2, 2, 2, 4, 9;
1, 1, 1, 1, 1, 2, 2, 2, 4, 10;
...
Array adjusted such that the rows represent base n and the columns m:
                       m
         1  2  3  4  5  6  7  8  9  10
         ------------------------------
base 1:  1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
base 2:  1, 2, 2, 3, 3, 3, 3, 4, 4, (4);
base 3:  1, 1, 2, 2, 2, 2, 2, 2, (3, 3);
base 4:  1, 1, 1, 2, 2, 2, 2, (2, 2, 2);
base 5:  1, 1, 1, 1, 2, 2, (2, 2, 2, 2);
base 6:  1, 1, 1, 1, 1, (2, 2, 2, 2, 2);
base 7:  1, 1, 1, 1, (1, 1, 2, 2, 2, 2);
base 8:  1, 1, 1, (1, 1, 1, 1, 2, 2, 2);
base 9:  1, 1, (1, 1, 1, 1, 1, 1, 2, 2);
base 10: 1, (1, 1, 1, 1, 1, 1, 1, 1, 1);
...
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.
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)

Mathematica

Table[Function[k, If[k == 1, m, IntegerLength[m, k]]][k - m + 1], {k, 13}, {m, k}] // Flatten (* Michael De Vlieger, Aug 31 2017 *)

Crossrefs

Cf. A162320.

Sequence in context: A133674 A348245 A215026 * A237594 A098666 A374443

Adjacent sequences: A162316 A162317 A162318 * A162320 A162321 A162322

Keyword

base,nonn,tabl

Author

Leroy Quet, Jul 01 2009

Status

approved