A164778 -id:A164778 - cp's OEIS frontend

A164772 Numbers n with property that average digit of n^2 is s=8.

313, 298327, 9893887, 197483417, 282753937, 314623583, 315432874, 706399164, 773303937, 894303633, 947047833, 948675387, 989938887, 994987437, 998398167, 8882454614, 8888194417, 8943142063, 9374913167, 9416989417, 9476286187, 9949260676, 9949823114

Offset: 1

Zak Seidov, Aug 26 2009

			a(1)=313 because 313^2=97969 and (9+7+9+6+9)/5=8.

Subsequence of A164817.

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).

More terms from David Radcliffe Aug 26 2009

a(16)-a(23) from Lars Blomberg, Apr 29 2013

A164773 Numbers n with property that average digit of n^2 is s=7.

Original entry on oeis.org

1667, 2167, 2383, 2387, 2437, 2563, 2567, 2617, 2626, 2824, 2828, 2963, 3143, 3157, 17313, 19437, 19917, 21417, 21633, 22083, 22113, 22293, 23214, 23622, 23664, 23874, 23937, 24207, 24228, 24267, 24417, 24474, 25824, 25836, 25863, 26067

Offset: 1

Zak Seidov, Aug 26 2009

			1667^2=2778889 and (2+7+7+8+8+8+9)/7=7
17313^2=299739969 and (2+9+9+7+3+9+9+6+9)/9=7.

Harvey P. Dale, Table of n, a(n) for n = 1..925

Subsequence of A164817.

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).

Select[Range[30000],Mean[IntegerDigits[#^2]]==7&] (* Harvey P. Dale, Feb 22 2013 *)

dsum(n)={my(s=0);while(n>9,s+=n%10;n\=10);s+n};
for(n=1,1e6,if(dsum(n^2)/#Str(n^2)==7,print1(n","))) \\ Charles R Greathouse IV, Nov 01 2009

A164770 Numbers k with the property that the average digit of k^2 is 2.

Original entry on oeis.org

145, 152, 179, 190, 251, 3182, 3190, 3199, 3245, 3290, 3335, 3362, 3380, 3470, 3479, 3496, 3550, 3649, 3650, 3749, 3821, 4001, 4010, 4100, 4495, 4496, 4540, 4550, 4585, 4595, 4639, 4649, 4810, 4820, 4910, 4990, 5701, 5710, 5755, 5800, 5900, 6350, 6404

Offset: 1

Zak Seidov, Aug 26 2009

There are 6368 such k's < 10^7: see link to 15217.html.

			145 is a term because 145^2 = 21025 and (2 + 1 + 0 + 2 + 5)/5 = 2.

Harvey P. Dale, Table of n, a(n) for n = 1..2500
Zak Seidov, Average digit of n^2 is 2.
Zak Seidov, Case with trailing zeros.

Subsequence of A164817.

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).

Select[Range[6500],Mean[IntegerDigits[#^2]]==2&] (* Harvey P. Dale, May 10 2021 *)

dsum(n)={my(s=0);while(n>9,s+=n%10;n\=10);s+n};
for(n=1,1e6,if(dsum(n^2)/#Str(n^2)==2,print1(n","))) \\ Charles R Greathouse IV, Nov 01 2009

Edited by Charles R Greathouse IV, Mar 23 2010

A164771 Numbers k such that the average digit of k^2 is 1.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Aug 26 2009

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.

Or: Numbers k such that k^2 is in A061384, i.e., square root of squares in A061384. - M. F. Hasler, Dec 05 2010

			1049 is a term because 1049^2 = 1100401 and (1 + 1 + 0 + 0 + 4 + 0 + 1)/7 = 1.

Subsequence of A164817.
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).
Cf. A007953, A055642, A061384.

Select[Range[50000],Mean[IntegerDigits[#^2]]==1&] (* Harvey P. Dale, Dec 15 2014 *)

{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

A055642(a(n)^2) = A007953(a(n)^2). - M. F. Hasler, Dec 05 2010

Terms up to a(117) checked with given PARI code by M. F. Hasler, Dec 05 2010

A164774 Numbers n with property that average digit of n^2 is s=5.

Original entry on oeis.org

8, 113, 122, 157, 158, 166, 176, 184, 193, 194, 212, 221, 238, 256, 257, 266, 274, 275, 283, 284, 292, 311, 3283, 3314, 3386, 3391, 3413, 3458, 3463, 3517, 3544, 3562, 3593, 3674, 3683, 3697, 3724, 3733, 3737, 3764, 3814, 3827, 3836, 3859, 3863, 3872, 3917

Offset: 1

Zak Seidov, Aug 26 2009

			8^2=64 and (6+4)/2=5
113^2=12769 and (1+2+7+6+9)/5=5.

Subsequence of A164817.

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).

dsum(n)={my(s=0);while(n>9,s+=n%10;n\=10);s+n};
for(n=1,1e6,if(dsum(n^2)/#Str(n^2)==5,print1(n","))) \\ Charles R Greathouse IV, Nov 01 2009

A164776 Numbers n with property that average digit of n^2 is s=4.

Original entry on oeis.org

2, 41, 58, 59, 68, 85, 95, 1027, 1034, 1036, 1072, 1081, 1088, 1108, 1124, 1135, 1144, 1153, 1169, 1232, 1234, 1243, 1252, 1259, 1268, 1277, 1279, 1295, 1297, 1306, 1315, 1331, 1340, 1342, 1349, 1358, 1360, 1369, 1385, 1394, 1405, 1423, 1441, 1448, 1457

Offset: 1

Zak Seidov, Aug 26 2009

			41^2 = 1681 and (1 + 6 + 8 + 1)/4 = 4
58^2 = 3364 and (3 + 3 + 6 + 4)/4 = 4.

Subsequence of A164817.

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).

dsum(n)={my(s=0);while(n>9,s+=n%10;n\=10);s+n};
for(n=1,1e6,if(dsum(n^2)/#Str(n^2)==4,print1(n","))) \\ Charles R Greathouse IV, Nov 01 2009

A164782 Numbers k with property that average digit of k^2 is 3.

Original entry on oeis.org

12, 15, 18, 21, 30, 330, 339, 345, 354, 360, 369, 375, 381, 399, 402, 405, 420, 429, 453, 459, 462, 465, 468, 471, 489, 492, 495, 498, 504, 540, 552, 555, 558, 561, 570, 579, 585, 639, 642, 645, 651, 660, 690, 708, 711, 720, 729, 735, 750, 780, 789, 795, 801

Offset: 1

Zak Seidov, Aug 26 2009

All terms are multiples of 3.

			a(1) = 12 because 12^2 = 144 and (1 + 4 + 4)/3 = 3.
a(53) = 801 because 801^2 = 641601 and (6 + 4 + 1 + 6 + 0 + 1)/6 = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A164817.

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).

Filtered([1..801],n->Sum(ListOfDigits(n^2))/Size(ListOfDigits(n^2))=3); # Muniru A Asiru, Nov 01 2018

filter:= proc(n) local L;
L:= convert(n^2,base,10);
convert(L,`+`)=3*nops(L)
end proc:
select(filter, [seq(i,i=3..1000,3)]); # Robert Israel, Nov 01 2018

s={};Do[If[3==Mean[IntegerDigits[n^2]],Print[n];AppendTo[s,n]],{n,3,1000,3}];s
Select[Range[1000],Mean[IntegerDigits[#^2]]==3&] (* Harvey P. Dale, Jan 13 2015 *)

A379602 a(n) is the least n-digit number whose square contains only digits greater than 5.

Original entry on oeis.org

3, 26, 264, 3114, 25824, 260167, 2639867, 25845676, 260147437, 2582245083, 25843178924, 258241744863, 2582010592114, 25825761924437, 258218875510676, 2581990857627114, 25820083014911063, 258199298347206526, 2581988959445543367, 25819892911624938937, 258198891881411585714

Offset: 1

Zhining Yang, Dec 27 2024

Exists for all n because A379603(n) does (see Formulas there). - Michael S. Branicky, Dec 30 2024

			a(3) = 264 because among all 3-digit numbers, 264 is the smallest whose square 69696 contains only digits greater than 5.

Cf. A053972, A053974, A058471, A164778, A164772, A164773, A164841, A291631, A379603.

f[m_] := For[k = Ceiling@Sqrt[100^m/15], k < 10^m - 1, k++, If[Min@IntegerDigits[k^2] > 5, Return[k];]]; Table[f[m], {m, 10}]

a(9) corrected and a(11) inserted by Michael S. Branicky, Dec 27 2024

More terms from Jinyuan Wang, Dec 27 2024

A379603 a(n) is the largest n-digit number whose square contains only digits greater than 5.

Original entry on oeis.org

3, 83, 937, 9833, 98336, 998333, 9994833, 99983333, 999939437, 9999833333, 99998333336, 999998333333, 9999983333336, 99999983333333, 999999833333336, 9999999833333333, 99999998333333336, 999999998333333333, 9999999983333333336, 99999999983333333333, 999999999833333333336

Offset: 1

Zhining Yang, Dec 27 2024

			a(3) = 937 because among all 3-digit numbers, 937 is the largest whose square 877969 contains only digits greater than 5.

Cf. A053972, A053974, A058471, A164778, A164772, A164773, A164841, A291631, A379602.

f[m_] := For[k = 10^m - 1, k > 10^(m - 1), k--, If[Min@IntegerDigits[k^2] > 5, Return[k];]];
Table[f[m], {m, 10}]

Conjecture: It appears that for all n >= 5,
a(2*n) = 100^n - (5*10^n + 1)/3, and
a(2*n + 1) = 10*a(2*n) + 6.

a(20)-a(21) from Jinyuan Wang, Dec 27 2024

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A164772 Numbers n with property that average digit of n^2 is s=8.

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A164773 Numbers n with property that average digit of n^2 is s=7.

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A164770 Numbers k with the property that the average digit of k^2 is 2.

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A164771 Numbers k such that the average digit of k^2 is 1.

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A164774 Numbers n with property that average digit of n^2 is s=5.

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A164776 Numbers n with property that average digit of n^2 is s=4.

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A164782 Numbers k with property that average digit of k^2 is 3.

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A379602 a(n) is the least n-digit number whose square contains only digits greater than 5.

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