A119511 -id:A119511 - cp's OEIS frontend

A119998 a(n) is the smallest positive integer whose square starts with (at least) n identical digits.

1, 15, 149, 2357, 10541, 57735, 745356, 1490712, 182574186, 298142397, 298142397, 1490711985, 1490711985, 10540925533895, 74535599249993, 105409255338946, 7453559924999299, 14907119849998598, 471404520791031683, 7453559924999298988

Offset: 1

Tanya Khovanova, Jul 26 2006

			Note that 15^2 = 225 and 149^2 = 22201.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A119511, A192934.

a(n) <= A119511(n). - R. J. Mathar, Aug 08 2006

a(n) = sqrt(A192934(n)). - Alois P. Heinz, Sep 02 2011

More terms from R. J. Mathar, Jul 29 2006

More terms from R. J. Mathar, Aug 08 2006

A132391 Numbers whose square starts with 4 identical digits.

Original entry on oeis.org

2357, 2582, 3334, 4714, 5774, 6667, 8165, 8819, 9428, 10541, 10542, 10543, 10544, 10545, 14907, 14908, 14909, 18257, 18258, 18259, 21081, 21082, 21083, 23570, 23571, 25819, 25820, 27888, 27889, 29813, 29814, 31622, 33332, 33333

Offset: 1

Jonathan Vos Post, Aug 29 2007

			Example: 2357^2 = 5555449.

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

Cf. A119887, A119511, A131573, A119866, A133183.

R:= NULL: count:= 0:
for d from 1 while count < 100 do
  for i from 1 to 9 do
    L:= i*1111*10^d;
    X:= [$ceil(sqrt(L)) .. floor(sqrt(L+10^d-1))];
    m:= nops(X);
    if m > 0 then
      count:= count+nops(X);
      R:= R, op(X);
    fi
od od:
R; # Robert Israel, Mar 12 2021

Select[Range[10, 50000], Length[Union[Take[IntegerDigits[ #^2], 4]]] == 1 & ]
(* or *)
(* Here's a more generic Mathematica program that calculates the first q terms of squares starting with n identical digits *)
n=4; q=30; t=Table[(10^n-1)*i/9, {i,1,9}]; u=Sqrt[Union[t,10*t]];
v=Sqrt[Union[t+1, 10*(t+1)]]; k=1; While[s=Sort[Flatten[Table[Union
[Table[Range[Ceiling[10^j*u[[i]]], f=10^j*v[[i]]; If[IntegerQ[f],
f=f-1]; Floor[f]], {i,1,18}]], {j,0,k}]]]; Length[s]Hans Havermann, Aug 30 2007 *)

def aupto(limit):
  alst = []
  for m in range(34, limit+1):
    if len(set(str(m*m)[:4])) == 1: alst.append(m)
  return alst
print(aupto(33333)) # Michael S. Branicky, Mar 12 2021

A119866 Numbers whose square starts with 5 identical digits.

Original entry on oeis.org

10541, 33334, 47141, 57735, 66667, 105409, 105410, 105411, 105412, 105413, 149071, 149072, 149073, 182574, 182575, 182576, 210818, 210819, 235702, 235703, 258198, 258199, 278886, 278887, 298141, 298142, 316227, 333332, 333333, 333334

Offset: 1

Zak Seidov, Aug 04 2006

			10541^2=111112681, 33334^2=1111155556.

Cf. A119511, A131573, A132391, A133183.

Select[ Range[ 100, 500000 ], Length[ Union[ Take[ IntegerDigits[ #^2 ], 5 ] ] ] == 1 & ] (* Jonathan Vos Post, Aug 29 2007 *)

A119887 Numbers whose square starts with 6 identical digits.

Original entry on oeis.org

57735, 333334, 471405, 577350, 666667, 745356, 881917, 942809, 1054093, 1054094, 1054095, 1054096, 1490712, 1490713, 1490714, 1825741, 1825742, 1825743, 2108185, 2108186, 2357022, 2357023, 2581988, 2581989, 2788866, 2788867

Offset: 1

Zak Seidov, Aug 04 2006

			57735^2=3333330225, 333334^2=111111555556.

Cf. A119511.

Select[ Range[ 1000, 5000000 ], Length[ Union[ Take[ IntegerDigits[ #^2 ], 6 ] ] ] == 1 & ] (* Jonathan Vos Post, Aug 29 2007 *)

A131699 Smallest number whose n-th power begins with precisely n identical digits (in base ten).

Original entry on oeis.org

1, 15, 322, 167, 6444, 32183, 7306, 225418, 6551032, 683405939, 7074698775, 26331754107, 844494314469, 11303028458639, 251188643150958, 93364101391902, 16114920282762613, 239390020079624346, 191165654339590395

Offset: 1

Jonathan Vos Post and Hans Havermann, Sep 15 2007

Main diagonal of array A[k,n] = n-th positive integer whose square (base 10) begins with k identical digits. M. F. Hasler points out that numbers whose squares start with 4 identical digits; numbers whose squares start with 5 identical digits; and numbers whose squares start with 6 identical digits; are already in the OEIS (along with A119511, A119998).
For the less stringent condition of the n-th power beginning with at least n identical digits, replace the numbers at indices {14,23,27,49,53} with:
1247955519394
2237770493401064693452
119060799886319434107761934
1389495494373137637129985217353011622113046714491
6489094571807720876517179893325894917102663447322282, respectively.

			a(1) = 1 because 1^1 = 1 begins with precisely 1 identical digit.
a(2) = 15 because 15^2 = 225 begins with precisely 2 identical digits.
a(3) = 322 because 322^3 = 33386248 begins with precisely 3 identical digits.
a(4) = 167 because 167^4 = 777796321 begins with precisely 4 identical digits.
a(5) = 6444 because 6444^5 = 11111627111310388224 begins with precisely 5 identical digits.

Hans Havermann, Table of n, a(n) for n=1..100

See A132392 for another version.

Cf. A119511, A119998, A131573.

a(n) = Min{k>0 such that k^n begins with precisely n identical leftmost digits (base ten)}.

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A192934 Smallest square number starting with (at least) n identical digits.

Original entry on oeis.org

1, 225, 22201, 5555449, 111112681, 3333330225, 555555566736, 2222222266944, 33333333393562596, 88888888888905609, 88888888888905609, 2222222222222640225, 2222222222222640225, 111111111111119590793871025, 5555555555555557064110500049, 11111111111111115805344390916

Offset: 1

J. Lowell, Aug 25 2011

Conjecture: total number of digits of a(n) is always 2*n-1.

This conjecture is false: length(a(6)) = 10 != 2*6-1 = 11.

			a(2) = 225 because none of 11, 22, 33, 44, 55, 66, 77, 88, 99, any integer from 110 to 119, or any integer from 220 to 224 is a square, but 15^2 = 225.

Alois P. Heinz, Table of n, a(n) for n = 1..150

Cf. A119998, A119511.

a(n) = {if (n == 1, return (1)); ok = 0; i = 1; while (! ok, i++; d = digits(i^2, 10); if (#d >= n, ok = 1; for (k = 2, n, if (d[k] != d[1], ok = 0; break;);););); return (i^2);} \\ Michel Marcus, Jun 14 2013

a(n) = A119998(n)^2.

More terms from Alois P. Heinz, Sep 02 2011

A346926 a(n) is the smallest positive integer whose square starts and ends with exactly n identical digits, and a(n) = 0 when there is no such integer.

Original entry on oeis.org

1, 88, 10538, 235700, 0, 57735000, 0, 14907120000, 0, 235702260400000, 0, 7453559925000000, 0, 105409255338950000000, 0, 10540925533894600000000, 0, 14907119849998598000000000, 0, 74535599249992989880000000000, 0, 210818510677891955466600000000000, 0

Offset: 1

Bernard Schott, Aug 07 2021

When a square ends in exactly three identical digits, these digits are necessarily 444 (A039685).
When a square ends with n > 3 identical digits, these last digits are necessarily 0's, and also this is only possible when n is even.
Differs from A174499 where only at least n identical digits are required.

			a(2) = 88 because 88^2 = 7744 starts with two 7's and ends with two 4's, and 88 is the smallest integer whose square starts and ends with exactly 2 identical digits.
a(4) = 235700 because 235700^2 = 55554490000 starts with four 5's and ends with four 0's, and 235700 is the smallest integer whose square starts and ends with exactly 4 identical digits.

Cf. A039685, A119511, A174499, A346774, A346892.

a(2*n+1) = 0 for n >= 2.

a(2*n) = A119511(2*n) * 10^n, for n >= 2.

A167712 a(n) = the smallest positive number, not ending in 0, whose square has a substring of exactly n identical digits.

Original entry on oeis.org

1, 12, 38, 1291, 10541, 57735, 364585, 1197219, 50820359, 169640142, 298142397, 4472135955, 1490711985, 2185812841434

Offset: 1

Zak Seidov, Nov 10 2009

			a(1)=1: 1^2=1 (1 one), a(1)=A119511(1)=A119998(1)
a(2)=12: 12^2=144 (2 fours)
a(3)=38: 38^2=1444 (3 fours)
a(4)=1291: 1291^2=1666681 (4 sixes)
a(5)=10541: 10541^2=111112681 (5 ones), a(5)=A119511(5)=A119998(5)
a(6)=57735: 57735^2=3333330225 (6 threes), a(6)=A119511(6)=A119998(6)
a(7)=364585: 364585^2=132922222225 (7 twos)
a(8)=1197219: 1197219^2=1433333333961 (8 threes)
a(9)=50820359: 50820359^2=2582708888888881 (9 eights)
a(10)=169640142: 169640142^2=28777777777780164 (10 sevens)
a(11)=298142397: 298142397^2=88888888888905609 (11 eights), a(11)=A119511(11)=A119998(11)
a(12)=4472135955: 4472135955^2=20000000000003762025 (12 zeros)
a(13)=1490711985: 1490711985^2=2222222222222640225 (13 twos), a(13)=A119511(13)=A119998(12,13).

Cf. A119511, A119998, A131573, A132391, A119866, A119887, A285056.

a[n_] := Block[{k=1}, While[Mod[k, 10] == 0 || !MemberQ[Length /@ Split[ IntegerDigits[ k^2]], n], k++]; k]; Array[a, 7] (* Giovanni Resta, Apr 11 2017 *)

a(14) from Giovanni Resta, Apr 11 2017

A174499 Smallest number whose square starts and ends with (at least) n identical digits.

Original entry on oeis.org

1, 88, 10538, 235700, 10541000, 57735000, 7453560000, 14907120000, 18257418600000, 29814239700000, 298142397000000, 1490711985000000, 14907119850000000, 105409255338950000000, 7453559924999300000000, 10540925533894600000000

Offset: 1

Michel Lagneau, Feb 22 2011

For n > 3 the last n identical digits are zeros. Proof:
For n = 3, the numbers a(n) == {0, 38, 100, 200, 300, 400, 462, 500, 538, 600, 700, 800, 900, 962} mod 1000, but for n = 4, if the suffix is different from zero, a(n) == {38, 462, 538, 962} mod 1000, and for d from [1..9], (d038)^2 <> 4444 (mod 10000), (d462)^2 <> 4444 (mod 10000), (d538)^2 <> 4444 (mod 10000), (d962)^2 <> 4444 (mod 10000).
Differs from A346926 where exactly n identical digits are required. - Bernard Schott, Aug 08 2021

			a(3) = 10538 because 10538^2 = 111049444 starts and ends in 3 identical digits.
a(5) = 10541000 because 10541000^2 = 111112681000000 starts with 5 identical digits and ends with 6 identical digits.

Cf. A119511, A119998, A186438, A186439, A346926.

with(numtheory):T:=array(1..100):p0:=10:for k from 2 to 10 do: id:= 0:for p
  from p0 to 100000000 while(id=0) do:n:=p^2:l:=length(n):n0:=n:for m from 1 to
  l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v :T[m]:=u:od:z:=0:for a from 1
  to k-1 do: if T[l]=T[l-a] and T[1]=T[1+a] then z:=z+1:else fi:od:if z=k-1 then
  print(p):id:=1:p0:=p:else fi:od:od:

For n > 3, a(n) = A119998(n)*10^q, q = floor(n+1)/2. [corrected by Bernard Schott, Aug 08 2021]

Name clarified and a(10) and a(12) corrected by Bernard Schott, Aug 08 2021

cp's OEIS Frontend

A119998 a(n) is the smallest positive integer whose square starts with (at least) n identical digits.

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A132391 Numbers whose square starts with 4 identical digits.

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A119866 Numbers whose square starts with 5 identical digits.

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A119887 Numbers whose square starts with 6 identical digits.

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Mathematica

A131699 Smallest number whose n-th power begins with precisely n identical digits (in base ten).

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A192934 Smallest square number starting with (at least) n identical digits.

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PARI

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A346926 a(n) is the smallest positive integer whose square starts and ends with exactly n identical digits, and a(n) = 0 when there is no such integer.

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A167712 a(n) = the smallest positive number, not ending in 0, whose square has a substring of exactly n identical digits.

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Mathematica

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A174499 Smallest number whose square starts and ends with (at least) n identical digits.

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