cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Zak Seidov, Aug 26 2009

Keywords

Examples

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

Crossrefs

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).

Extensions

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

Views

Author

Zak Seidov, Aug 26 2009

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

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

Views

Author

Zak Seidov, Aug 26 2009

Keywords

Comments

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

Examples

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

Links

Crossrefs

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).

Programs

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

Extensions

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

Views

Author

Zak Seidov, Aug 26 2009

Keywords

Comments

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

Examples

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

Crossrefs

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.

Programs

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

Formula

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

Extensions

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

Views

Author

Zak Seidov, Aug 26 2009

Keywords

Examples

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

Crossrefs

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).

Programs

  • PARI
    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

Views

Author

Zak Seidov, Aug 26 2009

Keywords

Examples

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

Crossrefs

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).

Programs

  • PARI
    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

A164778 Numbers n with property that average digit of n^2 is s=6.

Original entry on oeis.org

24, 27, 387, 417, 423, 447, 528, 537, 543, 624, 663, 678, 684, 687, 714, 768, 774, 786, 813, 816, 822, 828, 834, 864, 867, 876, 882, 888, 891, 894, 924, 927, 933, 936, 942, 948, 957, 963, 969, 972, 978, 984, 987, 993, 10386, 11313, 11772, 12114, 12186
Offset: 1

Views

Author

Zak Seidov, Aug 26 2009

Keywords

Comments

All terms are multiples of 3.

Examples

			24^2 = 576 and (5 + 7 + 6)/3 = 6
387^2 = 149769 and (1 + 4 + 9 + 7 + 6 + 9)/6 = 6.

Links

Crossrefs

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).

Programs

  • Mathematica
    s={};me=6;Do[If[me==Mean[IntegerDigits[n^2]],Print[n];AppendTo[s,n]],{n,3,10^4,3}];s
Select[3*Range[5000],Mean[IntegerDigits[#^2]]==6&] (* Harvey P. Dale, Aug 17 2014 *)
  • PARI
    dsum(n)={my(s=0);while(n>9,s+=n%10;n\=10);s+n};
forstep(n=3,1e6,3,if(dsum(n^2)/#Str(n^2)==6,print1(n","))) \\ Charles R Greathouse IV, Nov 01 2009

A164841 Numbers whose squares have a digit average greater than 8.

Original entry on oeis.org

3, 94863, 987917, 3162083, 29983327, 99477133, 99483667, 994927133, 2428989417, 2754991833, 2983284917, 2999833327, 3157196367, 9380293167, 9486778167, 28105157886, 31144643167, 31304790167, 31459487917, 31464263856, 94286790167, 99497231067, 244272388937
Offset: 1

Views

Author

Zak Seidov, Aug 28 2009

Keywords

Comments

There are just 13 terms < 32*10^8.
3 is the only term whose square has a digit average that is an integer.
The squares of the first few terms have digit averages 9, 8.1, 8.08333, 8.15385, 8.06667, 8.125, 8.125, 8.22222, 8.05263, 8.05263, 8.10526, 8.10526, 8.05263, ...
The sequence contains all numbers of the form floor(30*100^k - 10^k*5/3), k > 5. As of today, we know of only 9 numbers whose square has a digit mean above 8.25: 3, 707106074079263583, 943345110232670883, 94180040294109027313, 2976388751488907738914, 312713447088224669275583, 893241282627485818275387, 314610537013606681884298837387 and 9984988582817657883693383344833. - M. F. Hasler, Apr 11 and Apr 13 2023

Links

Crossrefs

Cf. A164771 (s=1), A164770 (s=2), A164782 (s=3), A164776 (s=4), A164774 (s=5), A164779 (s=6), A164773 (s=7), A164772 (s=8).
Cf. A164842 (s < 1).

Programs

  • PARI
    for(L=1,oo, for(n=sqrtint(10^(L-1)-1)+1,sqrtint(10^L-1), sumdigits(n^2) > 8*L && print1(n", "))) \\ M. F. Hasler, Apr 11 2023

Extensions

a(14)-a(23) from Lars Blomberg, Apr 29 2013

A164842 Numbers n with property that average digit of n^2 is less than 1.

Original entry on oeis.org

101, 1001, 10001, 50001, 100001, 100002, 100005, 100011, 100101, 100549, 101001, 110001, 114499, 200001, 316261, 375501, 500001, 1000001, 1000002, 1000005, 1000011, 1000101, 1001001, 1010001, 1100001, 2000001, 5000001, 5000002
Offset: 1

Views

Author

Zak Seidov, Aug 28 2009

Keywords

Comments

No trailing zeros in n.
Other terms < 10^8:
5000011,5000101,5001001,5010001,5100001,5500001,6000001,10000001,10000002,
10000005,10000006,10000011,10000015,10000051,10000055,10000101,10000105,
10000501,10001001,10001005,10005001,10010001,10010005,10050001,10100001,
10100005,10500001,11000001,11000005,15000001,17320511,20000001,20000005,
44989999,48989999,50000001,50000002,50000011,50000101,50001001,50010001,
50100001,51000001,55000001,60000001.

Crossrefs

Cf. A164770 (s=2), A164771 (s=1), A164772 (s=8), A164773 (s=7), A164774 (s=5). A164776 (s=4). A164779 (s=6), A164782(s=3), A164841(s<8).

Extensions

Edited by Charles R Greathouse IV, Mar 23 2010
Showing 1-9 of 9 results.

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