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%I #18 Jul 15 2018 12:34:12
%S 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,
%T 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,
%U 0,14,0,0,0,0,52,0,0,499,0,0,0,0,0,25,191,1281,2658,2282,705,65,0,0
%N Table read by rows: T(n,k), 0 <= k <= 9, is the number of n-digit squares whose average digit is exactly k.
%C The only square whose average digit is 0 is the 1-digit number 0^2 = 0.
%C The only square whose average digit is 9 is the 1-digit number 3^2 = 9.
%C 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
%C - if k = 1, 4, or 7, then n mod 9 = 0, 1, 4, or 7;
%C - if k = 2, 5, or 8, then n mod 9 = 0, 2, 5, or 8;
%C - if k = 3 or 6, then n mod 9 = 0, 3, or 6.
%C In this table, each possible combination of a value of k and a value of n mod 9 is identified with an asterisk (*):
%C .
%C n mod 9
%C .
%C 0 1 2 3 4 5 6 7 8
%C +----------------------------------
%C 1 | * * * *
%C |
%C 2 | * * * *
%C |
%C 3 | * * *
%C |
%C 4 | * * * *
%C k |
%C 5 | * * * *
%C |
%C 6 | * * *
%C |
%C 7 | * * * *
%C |
%C 8 | * * * *
%C .
%C 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.
%H Jon E. Schoenfield, <a href="/A316480/b316480.txt">Table of n, a(n) for n = 1..190</a>
%e Table begins
%e n\k| 0 1 2 3 4 5 6 7 8 9
%e ---+---------------------------------------------------------
%e 1 | 1 1 0 0 1 0 0 0 0 1
%e 2 | 0 0 0 0 0 1 0 0 0 0
%e 3 | 0 0 0 5 0 0 2 0 0 0
%e 4 | 0 0 0 0 6 0 0 0 0 0
%e 5 | 0 0 5 0 0 21 0 0 1 0
%e 6 | 0 0 0 57 0 0 42 0 0 0
%e 7 | 0 2 0 0 192 0 0 14 0 0
%e 8 | 0 0 52 0 0 499 0 0 0 0
%e 9 | 0 25 191 1281 2658 2282 705 65 0 0
%e 10 | 0 12 0 0 5308 0 0 93 0 0
%e 11 | 0 0 548 0 0 13597 0 0 1 0
%e 12 | 0 0 0 23310 0 0 12871 0 0 0
%e 13 | 0 77 0 0 143724 0 0 753 0 0
%e 14 | 0 0 5572 0 0 360720 0 0 1 0
%e 15 | 0 0 0 449170 0 0 239403 0 0 0
%e 16 | 0 102 0 0 3990950 0 0 6029 0 0
%e 17 | 0 0 51977 0 0 9994767 0 0 4 0
%e 18 | 0 417 157382 8665925 55115308 45351595 4568205 36552 8 0
%t 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 *)
%Y Cf. A004159, A069711.
%Y Cf. A316481-A316488 (Squares whose arithmetic mean of digits is k, for k=1..8).
%K nonn,tabf,base
%O 1,24
%A _Jon E. Schoenfield_, Jul 04 2018
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