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

1,24

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.

Links

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

Example

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

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 *)

Crossrefs

Cf. A004159, A069711.

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

Sequence in context: A331039 A171915 A287703 * A099224 A136598 A193524

Adjacent sequences: A316477 A316478 A316479 * A316481 A316482 A316483

Keyword

nonn,tabf,base

Author

Jon E. Schoenfield, Jul 04 2018

Status

approved