A316480 Table read by rows: T(n,k), 0 <= k <= 9, is the number of n-digit squares whose average digit is exactly k.

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 21, 0, 0, 1, 0, 0, 0, 0, 57, 0, 0, 42, 0, 0, 0, 0, 2, 0, 0, 192, 0, 0, 14, 0, 0, 0, 0, 52, 0, 0, 499, 0, 0, 0, 0, 0, 25, 191, 1281, 2658, 2282, 705, 65, 0, 0

Offset: 1

Author

Jon E. Schoenfield, Jul 04 2018

Keywords

• nonn
• tabf
• base

Comments

The only square whose average digit is 0 is the 1-digit number 0^2 = 0.

The only square whose average digit is 9 is the 1-digit number 3^2 = 9.

Suppose m^2 is an n-digit number whose average digit is an integer k, i.e., digitsum(m^2) = n*k. Since digitsum(m^2) mod 9 = 0, 1, 4, or 7 (cf. A004159), it follows that

- if k = 1, 4, or 7, then n mod 9 = 0, 1, 4, or 7;

- if k = 2, 5, or 8, then n mod 9 = 0, 2, 5, or 8;

- if k = 3 or 6, then n mod 9 = 0, 3, or 6.

In this table, each possible combination of a value of k and a value of n mod 9 is identified with an asterisk (*):

.

n mod 9

.

0 1 2 3 4 5 6 7 8

+----------------------------------

1 | * * * *

|

2 | * * * *

|

3 | * * *

|

4 | * * * *

k |

5 | * * * *

|

6 | * * *

|

7 | * * * *

|

8 | * * * *

.

Not surprisingly, among the values k=1..8, the value of k that occurs least frequently as the average digit of a square is 8.

Examples

			Table begins
  n\k| 0   1      2       3        4        5       6     7 8 9
  ---+---------------------------------------------------------
   1 | 1   1      0       0        1        0       0     0 0 1
   2 | 0   0      0       0        0        1       0     0 0 0
   3 | 0   0      0       5        0        0       2     0 0 0
   4 | 0   0      0       0        6        0       0     0 0 0
   5 | 0   0      5       0        0       21       0     0 1 0
   6 | 0   0      0      57        0        0      42     0 0 0
   7 | 0   2      0       0      192        0       0    14 0 0
   8 | 0   0     52       0        0      499       0     0 0 0
   9 | 0  25    191    1281     2658     2282     705    65 0 0
  10 | 0  12      0       0     5308        0       0    93 0 0
  11 | 0   0    548       0        0    13597       0     0 1 0
  12 | 0   0      0   23310        0        0   12871     0 0 0
  13 | 0  77      0       0   143724        0       0   753 0 0
  14 | 0   0   5572       0        0   360720       0     0 1 0
  15 | 0   0      0  449170        0        0  239403     0 0 0
  16 | 0 102      0       0  3990950        0       0  6029 0 0
  17 | 0   0  51977       0        0  9994767       0     0 4 0
  18 | 0 417 157382 8665925 55115308 45351595 4568205 36552 8 0
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..190

Crossrefs

Cf. A004159, A069711.

Cf. A316481-A316488 (Squares whose arithmetic mean of digits is k, for k=1..8).

Programs

Mathematica

Block[{nn = 9, s}, s = MapAt[Prepend[#, 0] &, Map[Mean@ IntegerDigits[#] &, SplitBy[Range[10^(nn/2)]^2, IntegerLength], {2}], 1]; Table[Count[s[[n]], k], {n, nn}, {k, 0, 9}]] // Flatten (* Michael De Vlieger, Jul 06 2018 *)