A164841 -id:A164841 - cp's OEIS frontend

A370522 a(n) is the least n-digit number whose square has the maximum sum of digits (A348300(n)).

7, 83, 836, 8937, 94863, 987917, 9893887, 99477133, 994927133, 9380293167, 99497231067, 926174913167, 9892825177313, 89324067192437, 943291047332683, 9949874270443813, 83066231922477313, 707106074079263583, 9429681807356492126, 94180040294109027313, 888142995231510436417, 8882505274864168010583

Offset: 1

Zhining Yang, Feb 21 2024

a(n) is the last n-digit term in A067179.

As the last two of the only nine known numbers whose square has a digit mean above 8.25 (see A164841), there is a high probability that a(30)=314610537013606681884298837387 and a(31)=9984988582817657883693383344833.

			a(3) = 836 because among all 3-digit numbers, 836 is the smallest whose square 698896 has the maximum sum of digits, 46 = A348300(3).

Zhining Yang, Table of n, a(n) for n = 1..24 (terms 11..24 from Zhao Hui Du)

Cf. A004159, A067179, A348300, A348303.

A348300={13,31,46,63,81,97,112,130,148,162,180};
A370522[n_]:=Do[If[Total@IntegerDigits[k^2]==A348300[[n]],Return[k];],{k,10^(n-1),10^n-1}];
Table[A370522[n],{n,8}]

def A370522(n):
    A348300=[0,13,31,46,63,81,97,112,130,148,162,180]
    for k in range(10**(n-1), 10**n):
        if sum(int(d) for d in str(k**2))==A348300[n]:
            return(k)
print([A370522(n) for n in range(1,9)])

a(11)-a(24) from Zhao Hui Du, Feb 23 2024

A360803 Numbers whose squares have a digit average of 8 or more.

Original entry on oeis.org

3, 313, 94863, 298327, 987917, 3162083, 9893887, 29983327, 99477133, 99483667, 197483417, 282753937, 314623583, 315432874, 706399164, 773303937, 894303633, 947047833, 948675387, 989938887, 994927133, 994987437, 998398167, 2428989417, 2754991833, 2983284917, 2999833327

Offset: 1

Dmitry Kamenetsky, Feb 21 2023

This sequence is infinite. For example, numbers floor(30*100^k - (5/3)*10^k) beginning with 2 followed by k 9s, followed by 8 and k 3s, have a square whose digit average converges to (but never equals) 8.25. [Corrected and formula added by M. F. Hasler, Apr 11 2023]
Only a few examples are known whose square has a digit average of 8.25 and above: 3^2 = 9, 707106074079263583^2 = 499998999999788997978888999589997889 (digit average 8.25), 94180040294109027313^2 = 8869879989799999999898984986998979999969 (digit average 8.275).
This is the union of A164772 (digit average = 8) and A164841 (digit average > 8). - M. F. Hasler, Apr 11 2023

			94863 is in the sequence, because 94863^2 = 8998988769, which has a digit average of 8.1 >= 8.

Michael S. Branicky, Python program for OEIS A360822 and searching for squares with < given number of digits < 8
Matthieu Dufour and Silvia Heubach, Squares with Large Digit Average, 2020.
Dmitry Kamenetsky, The first 68 terms and their digit average

Cf. A004159, A185679.

Cf. A164772 (digit average = 8), A164841 (digit average > 8).

isok(k) = my(d=digits(k^2)); vecsum(d)/#d >= 8; \\ Michel Marcus, Feb 22 2023

def ok(n): d = list(map(int, str(n**2))); return sum(d) >= 8*len(d)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Feb 22 2023

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

Zak Seidov, Aug 28 2009

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.

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).

Edited by Charles R Greathouse IV, Mar 23 2010

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

A362264 Numbers > 9 with increasingly large digit average of their square, in base 10.

Original entry on oeis.org

10, 11, 12, 13, 17, 63, 83, 313, 94863, 3162083, 994927133

Offset: 0

M. F. Hasler, Apr 13 2023

The single-digit number 3, whose square is 9, has the highest possible digit average, therefore this "trivial solution" is excluded. However, the sequence could be defined as "numbers > 3 ..." in which case it would start 4, 6, 7, 63, ... see examples.
It is conjectured but not known that there are only finitely many numbers whose square has a digit average above 8.3.
Can it be proved or disproved that all terms > 17 end in a digit 3?
Next terms might be 707106074079263583 (da = 8.25) and 94180040294109027313 (da = 8.275), but there might be other terms in between.

			The respective digit averages are:
   n  |    a(n)   |       a(n)^2     | #digits | sum(digits) | digit average
  ----+-----------+------------------+---------+-------------+------------------
   -  |      4    |          16      |    2    |       7     |    7/2 = 3.5
   -  |      6    |          36      |    2    |       9     |    9/2 = 4.5
   -  |      7    |          49      |    2    |      13     |   13/2 = 6.5
   0  |     10    |         100      |    3    |       1     |    1/3 = 0.333...
   1  |     11    |         121      |    3    |       4     |    4/3 = 1.333...
   2  |     12    |         144      |    3    |       9     |     3  = 3.0
   3  |     13    |         169      |    3    |      16     |   16/3 = 3.333...
   4  |     17    |         289      |    3    |      19     |   19/3 = 6.333...
   5  |     63    |        3969      |    4    |      27     |   27/4 = 6.75
   6  |     83    |        6889      |    4    |      31     |   31/4 = 7.75
   7  |    313    |       97969      |    5    |      40     |     8  = 8.0
   8  |   94863   |     8998988769   |   10    |      81     |  81/10 = 8.1
   9  |  3162083  |   9998768898889  |   13    |     106     | 106/13 = 8.15...
  10  | 994927133 |989879999979599689|   18    |     148     |   74/9 = 8.222...

Cf. A164841, A068947, A068809.

m=0; for(k=10,oo, vecsum(d=digits(k^2))>m*#d && !print1(k", ") && m=vecsum(d)/#d)

cp's OEIS Frontend

A370522 a(n) is the least n-digit number whose square has the maximum sum of digits (A348300(n)).

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A360803 Numbers whose squares have a digit average of 8 or more.

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

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

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A362264 Numbers > 9 with increasingly large digit average of their square, in base 10.

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