A186438 -id:A186438 - cp's OEIS frontend

A305719 Numbers whose squares have the same first and last digits.

1, 2, 3, 11, 22, 26, 39, 41, 68, 75, 97, 101, 109, 111, 119, 121, 129, 131, 139, 141, 202, 208, 212, 218, 222, 225, 235, 246, 254, 256, 264, 303, 307, 313, 319, 321, 329, 331, 339, 341, 349, 351, 359, 361, 369, 371, 379, 381, 389, 391, 399, 401, 409, 411, 419, 421, 429, 431, 439, 441, 638

Offset: 1

Neville Holmes, Jun 08 2018

			For k = 11, k^2 = 121;
for k = 26, k^2 = 676.

Cf. A227858, A116499, A116501.

Cf. A123912, A186438.

Select[Range[638], (d = IntegerDigits[#^2]; d[[1]] == d[[-1]]) &] (* Giovanni Resta, Jun 25 2018 *)

for(n=1, 10^3, my(d=digits(n^2)); if( d[1]==d[#d], print1(n,", "))); \\ Joerg Arndt, Jun 10 2018

def ok(n): s = str(n*n); return s[0] == s[-1]
print(list(filter(ok, range(1, 639)))) # Michael S. Branicky, Jul 16 2021

A346774 Numbers whose square starts and ends with exactly 2 identical digits.

Original entry on oeis.org

88, 150, 210, 212, 338, 340, 470, 580, 670, 880, 940, 1050, 1060, 1062, 1070, 1080, 1088, 1090, 1488, 1510, 1512, 1820, 1830, 1838, 1840, 2110, 2112, 2120, 2350, 2360, 2362, 2570, 2580, 2588, 2780, 2790, 2970, 3150, 3160, 3320, 3330, 3350, 3360, 3362, 3370, 3380, 3388, 3390, 3410

Offset: 1

Bernard Schott, Aug 03 2021

The terminal digits are 00 or 44.

			150 is a term because 150^2 = 22500.
212 is a term because 212^2 = 44944 (smallest square with 2 times two 4's).
2788 is not a term because 2788^2 = 7772944.

Subsequence of A346678.

Cf. A123912, A186438, A186439.

Select[Range[32, 3500], (d = IntegerDigits[#^2])[[1]] == d[[2]] != d[[3]] && d[[-1]] == d[[-2]] != d[[-3]] &] (* Amiram Eldar, Aug 03 2021 *)

def ok(n):
    s = str(n*n)
    if len(s) < 4: return False # there are no ok squares with < 4 digits
    return s[0] == s[1] != s[2] and s[-1] == s[-2] != s[-3]
print(list(filter(ok, range(3411)))) # Michael S. Branicky, Aug 03 2021

A186439 Numbers whose squares end in three identical digits.

Original entry on oeis.org

38, 100, 200, 300, 400, 462, 500, 538, 600, 700, 800, 900, 962, 1000, 1038, 1100, 1200, 1300, 1400, 1462, 1500, 1538, 1600, 1700, 1800, 1900, 1962, 2000, 2038, 2100, 2200, 2300, 2400, 2462, 2500, 2538, 2600, 2700, 2800, 2900, 2962, 3000, 3038, 3100, 3200, 3300, 3400, 3462

Offset: 1

Michel Lagneau, Feb 21 2011

The three ending digits of a(n)^2 are 000 or 444.

n is in the sequence iff either n == 0 mod 100 or n == (+/-)38 mod 500. - Robert Israel, Jul 03 2014

			462 is in the sequence because 462^2 = 213444.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A016742 (even squares), A186438.
Cf. A039685, A047336.
Cf. A346678.

with(numtheory):T:=array(1..10):for p from 1 to 10000 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:if T[1]=T[2]
  and T[1]=T[3] then printf(`%d, `,p):else fi:od:
# second Maple program:
a:= proc(n) local m, r;
      r:= 1+ irem(n-1, 7, 'm');
      [38, 100, 200, 300, 400, 462, 500][r] +500*m
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 24 2011

Select[Range[11,10000],Mod[PowerMod[#,2,1000],111]==0&] (* Zak Seidov, Feb 23 2011 *)

for(n=11,10000,if((n^2%1000)%111==0,print1(n", "))) \\ Zak Seidov, Feb 23 2011

Vec(2*x*(19*x^2 +12*x +19)*(x^4 +x^3 +x^2 +x +1)/((x -1)^2*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)) + O(x^100)) \\ Colin Barker, Jul 03 2014

def ok(n): s = str(n*n); return len(s) > 2 and s[-1] == s[-2] == s[-3]
print(list(filter(ok, range(3463)))) # Michael S. Branicky, Jul 29 2021

a(A047336(n)) = A039685(n). - Bruno Berselli, Feb 22 2011
a(n) = a(n-7) + 500 for n > 7. - Zak Seidov and Bruno Berselli, Feb 23 2011
G.f.: 2*x*(19*x^2 +12*x +19)*(x^4 +x^3 +x^2 +x +1) / ((x -1)^2*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jul 03 2014

A346678 Positive numbers whose squares end in exactly two identical digits.

Original entry on oeis.org

10, 12, 20, 30, 40, 50, 60, 62, 70, 80, 88, 90, 110, 112, 120, 130, 138, 140, 150, 160, 162, 170, 180, 188, 190, 210, 212, 220, 230, 238, 240, 250, 260, 262, 270, 280, 288, 290, 310, 312, 320, 330, 338, 340, 350, 360, 362, 370, 380, 388, 390, 410, 412, 420, 430, 438, 440, 450, 460

Offset: 1

Bernard Schott, Jul 29 2021

When a square ends in exactly two identical digits, these digits are necessarily 00 or 44, so all terms are even.
The numbers are of the form: 10*floor((10*k-1)/9), k > 0, or, 50*floor((10*k-1)/9) +- 38, k > 0.
Equivalently: m is in the sequence iff either (m == 0 (mod 10) and m <> 0 (mod 100)) or (m == +- 38 (mod 50) and m <> +- 38 (mod 500)).

			12 is in the sequence because 12^2 = 144 ends in two 4's.
20 is in the sequence because 20^2 = 400 ends in two 0's.
38 is not in the sequence because 38^2 = 1444 ends in three 4's.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Equals A186438 \ A186439.

Supersequence of A346774.

Select[Range[10, 460], (d = IntegerDigits[#^2])[[-1]] == d[[-2]] != d[[-3]] &] (* Amiram Eldar, Jul 29 2021 *)

def ok(n): s = str(n*n); return len(s) > 2 and s[-1] == s[-2] != s[-3]
print(list(filter(ok, range(461)))) # Michael S. Branicky, Jul 29 2021

a(n+63) = a(n) + 500.

A346812 Positive numbers whose square starts with exactly 2 identical digits.

Original entry on oeis.org

15, 21, 34, 47, 58, 67, 88, 94, 105, 106, 107, 108, 109, 150, 151, 182, 183, 184, 210, 211, 212, 235, 236, 257, 258, 278, 279, 297, 315, 316, 332, 333, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 470, 471, 473, 474, 475, 476, 477, 478, 479, 575, 576, 577, 578, 579, 580, 581

Offset: 1

Bernard Schott, Aug 05 2021

If m is a term, then 10*m is another term.

			34 is a term because 34^2 = 1156.
149 is not a term because 149^2 = 22201.

Subsequence of A123912.
A346774 is a subsequence.
Cf. A186438, A186439, A346678 (similar, with "ends").

Select[Range[10, 600], (d = IntegerDigits[#^2])[[1]] == d[[2]] != d[[3]] &] (* Amiram Eldar, Aug 05 2021 *)

isok(m) = my(d=digits(m^2)); (#d > 2) && (d[2] == d[1]) && (d[3] != d[2]); \\ Michel Marcus, Aug 05 2021

def ok(n): s = str(n*n); return len(s) > 2 and s[0] == s[1] != s[2]
print(list(filter(ok, range(582)))) # Michael S. Branicky, Aug 05 2021

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

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A305719 Numbers whose squares have the same first and last digits.

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A346774 Numbers whose square starts and ends with exactly 2 identical digits.

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A186439 Numbers whose squares end in three identical digits.

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A346678 Positive numbers whose squares end in exactly two identical digits.

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A346812 Positive numbers whose square starts with exactly 2 identical digits.

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

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