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%I #26 Apr 11 2023 19:22:33
%S 1,1049,1490,10002,10005,10011,10020,10050,10101,10110,10149,10200,
%T 10500,11001,11010,11100,11490,12000,14499,15000,17610,18000,20001,
%U 20010,20100,21000,24900,30000,33200,35000,36100,44900,44990,45100
%N Numbers k such that the average digit of k^2 is 1.
%C There are 117 such n's < 10^7: 1, 1049, 1490, 10002, 10005, 10011, 10020, 10050, 10101, 10110, 10149, 10200, 10500, 11001, 11010, 11100, 11490, 12000, 14499, 15000, 17610, 18000, 20001, 20010, 20100, 21000, 24900, 30000, 33200, 35000, 36100, 44900, 44990, 45100, 46000, 54800, 55000, 64900, 71000, 80000, 1000006, 1000015, 1000051, 1000055, 1000060, 1000105, 1000150, 1000501, 1000510, 1000550, 1000600, 1001005, 1001050, 1001500, 1005001, 1005010, 1005100, 1005500, 1006000, 1006490, 1009951, 1010005, 1010050, 1010149, 1010500, 1011490, 1015000, 1024900, 1050001, 1050010, 1050100, 1051000, 1055000, 1060000, 1064900, 1095500, 1096000, 1100005, 1100050, 1100500, 1105000, 1114900, 1145000, 1150000, 1190000, 1224749, 1244990, 1249000, 1414249, 1415000, 1420000, 1424900, 1429000, 1451000, 1460000, 1484251, 1500001, 1500010, 1500100, 1501000, 1510000, 1550000, 1600000, 1735000, 1739000, 1789000, 1820000, 2000005, 2000050, 2000500, 2005000, 2050000, 2239000, 2261000, 2450000, 2500000, 2900000.
%C Or: Numbers k such that k^2 is in A061384, i.e., square root of squares in A061384. - _M. F. Hasler_, Dec 05 2010
%F A055642(a(n)^2) = A007953(a(n)^2). - _M. F. Hasler_, Dec 05 2010
%e 1049 is a term because 1049^2 = 1100401 and (1 + 1 + 0 + 0 + 4 + 0 + 1)/7 = 1.
%t Select[Range[50000],Mean[IntegerDigits[#^2]]==1&] (* _Harvey P. Dale_, Dec 15 2014 *)
%o (PARI) {for(d=1,9, for(n=sqrtint(10^(d-1)-1)+1, sqrtint(10^d-1), my(n2=divrem(n^2,10)); sum( k=2,d, (n2=divrem(n2[1],10))[2],n2[2])/d==1 & print1(n",")))} \\ _M. F. Hasler_, Dec 05 2010
%Y Subsequence of A164817.
%Y Average of digits of n^2 = s: A164771 (s=1), A164770 (s=2), A164782 (s=3), A164776 (s=4), A164774 (s=5), A164778 (s=6), A164773 (s=7), A164772 (s=8).
%Y Cf. A007953, A055642, A061384.
%K base,nonn
%O 1,2
%A _Zak Seidov_, Aug 26 2009
%E Terms up to a(117) checked with given PARI code by _M. F. Hasler_, Dec 05 2010
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