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A316480 Table read by rows: T(n,k), 0 <= k <= 9, is the number of n-digit squares whose average digit is exactly k. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 15 08:18 EDT 2021. Contains 342977 sequences. (Running on oeis4.)