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Positive numbers whose square starts with exactly 3 identical digits.
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149, 298, 334, 472, 667, 745, 882, 1054, 1055, 1056, 1057, 1058, 1490, 1491, 1492, 1493, 1825, 1826, 1827, 2108, 2109, 2356, 2581, 2788, 2789, 2980, 2981, 3161, 3162, 3332, 3333, 3335, 3336, 3337, 3338, 3339, 3340, 3341, 3342, 3343, 3344, 3345, 3346
COMMENTS
If m is a term 10*m is another term.
2357 is the first term of A131573 that is not in this sequence (see Example section), the next ones are 2582, 3334, ...
EXAMPLE
149 is a term because 149^2 = 22201 starts with three 2's.
2357 is not a term because 2357^2 = 5555449 starts with four 5's.
MATHEMATICA
Select[Range[32, 3350], (d = IntegerDigits[#^2])[[1]] == d[[2]] == d[[3]] != d[[4]] &] (* Amiram Eldar, Aug 06 2021 *)
PROG
(Python)
def ok(n): s = str(n*n); return len(s) > 3 and s[0] == s[1] == s[2] != s[3]
Numbers whose square starts with exactly 4 identical digits.
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2
2357, 2582, 3334, 4714, 5774, 6667, 8165, 8819, 9428, 10542, 10543, 10544, 10545, 14907, 14908, 14909, 18257, 18258, 18259, 21081, 21082, 21083, 23570, 23571, 25819, 25820, 27888, 27889, 29813, 29814, 31622, 33332, 33333, 33335, 33336, 33337, 33338, 33339, 33340, 33341, 33342
COMMENTS
If m is a term, 10*m is another term.
Differs from A132391 where only at least 4 identical digits are required; indeed, 10541 is the first term of A132391 that is not in this sequence (see Example section), the next one is 33346.
EXAMPLE
2357 is a term because 2357^2 = 5555449 starts with four 5's.
10541 is not a term because 10541^2 = 111112681 starts with five 1's.
MATHEMATICA
q[n_] := SameQ @@ (d = IntegerDigits[n^2])[[1 ;; 4]] && d[[5]] != d[[1]]; Select[Range[100, 33350], q] (* Amiram Eldar, Aug 08 2021 *)
PROG
(Python)
def ok(n):
s = str(n*n)
return len(s) > 4 and s[0] == s[1] == s[2] == s[3] != s[4]
Numbers whose square starts with 4 identical digits.
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4
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
EXAMPLE
Example: 2357^2 = 5555449.
MAPLE
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:
MATHEMATICA
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]<q, k++ ]; Take[s, q]
PROG
(Python)
def aupto(limit):
alst = []
for m in range(34, limit+1):
if len(set(str(m*m)[:4])) == 1: alst.append(m)
return alst
Smallest number whose n-th power begins with precisely n identical digits (in base ten).
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1, 15, 322, 167, 6444, 32183, 7306, 225418, 6551032, 683405939, 7074698775, 26331754107, 844494314469, 11303028458639, 251188643150958, 93364101391902, 16114920282762613, 239390020079624346, 191165654339590395
COMMENTS
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:
14 1247955519394
23 2237770493401064693452
27 119060799886319434107761934
49 1389495494373137637129985217353011622113046714491
53 6489094571807720876517179893325894917102663447322282, respectively.
FORMULA
a(n) = Min{k>0 such that k^n begins with precisely n identical leftmost digits (base ten)}.
EXAMPLE
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.
a(n) = the smallest positive number, not ending in 0, whose square has a substring of exactly n identical digits.
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1, 12, 38, 1291, 10541, 57735, 364585, 1197219, 50820359, 169640142, 298142397, 4472135955, 1490711985, 2185812841434
EXAMPLE
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(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).
MATHEMATICA
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 *)
Numbers whose square starts with 5 identical digits.
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2
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
EXAMPLE
10541^2=111112681, 33334^2=1111155556.
MATHEMATICA
Select[ Range[ 100, 500000 ], Length[ Union[ Take[ IntegerDigits[ #^2 ], 5 ] ] ] == 1 & ] (Jonathan Vos Post, Aug 29 2007)
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