A346774 -id:A346774 - cp's OEIS frontend

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

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.

A346892 Numbers whose square starts and ends with exactly 3 identical digits.

Original entry on oeis.org

10538, 33462, 99962, 105462, 105538, 149038, 182538, 298038, 333538, 333962, 334038, 334462, 334538, 471538, 471962, 472038, 577462, 577538, 666462, 666538, 666962, 667038, 745038, 745462, 745538, 816538, 881538, 881962, 882038, 942462, 942538, 999538, 1053962, 1054038, 1054538, 1054962

Offset: 1

Bernard Schott, Aug 06 2021

The terminal digits of the square of terms are necessarily 444.

The last 3 digits of terms are either 038, 462, 538 or 962. - Chai Wah Wu, Oct 02 2021

			10538 is a term because 10538^2 = 111049444
666462 = A348832(1) is a term because 666462^2 = 444171597444, the smallest square that starts with exactly three 4's and ends also with three 4's.
105462 is a term because 105462^2 = 11122233444 (see A079035).
74538 is not a term because 74538^2 = 5555913444 with four starting 5's.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Intersection of A039685 and A346891.
Cf. A346774 (similar, with 2 identical digits).
A348832 is a subsequence.

Select[Range[10^3, 10^6], (d = IntegerDigits[#^2])[[1]] == d[[2]] == d[[3]] != d[[4]] && d[[-1]] == d[[-2]] == d[[-3]] != d[[-4]] &] (* Amiram Eldar, Aug 06 2021 *)

def ok(n):
    s = str(n*n)
    if len(s) < 4: return False
    return s[0] == s[1] == s[2] != s[3] and s[-1] == s[-2] == s[-3] != s[-4]
print(list(filter(ok, range(10**6)))) # Michael S. Branicky, Aug 06 2021

A346892_list = [1000*n+d for n in range(10**6) for d in [38,462,538,962] if (lambda x:x[0]==x[1]==x[2]!=x[3])(str((1000*n+d)**2))] # Chai Wah Wu, Oct 02 2021

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

A346942 Numbers whose square starts and ends with exactly 4 identical digits.

Original entry on oeis.org

235700, 258200, 333400, 471400, 577400, 666700, 816500, 881900, 942800, 1054200, 1054300, 1054400, 1054500, 1490700, 1490800, 1490900, 1825700, 1825800, 1825900, 2108100, 2108200, 2108300, 2357100, 2581900, 2788800, 2788900, 2981300, 2981400, 3162200, 3333200, 3333300

Offset: 1

Bernard Schott, Aug 08 2021

Terms are equal to 100 times the primitive terms of A346940, those that have no trailing zero in decimal representation, hence all terms end with exactly 00.

			258200 is a term because 258200^2 = 66667240000 starts with four 6's and ends with four 0's.
3334700 is not a term because 3334700^2 = 1111155560000 starts with five 1's (and ends with four 0's).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Numbers whose square '....' with exactly k identical digits:
  ---------------------------------------------------------------------------
  |    k  \'....'|    starts     |       ends         |   starts and ends   |
  ---------------------------------------------------------------------------
  |     k = 2    |    A346812    |      A346678       |       A346774       |
  |     k = 3    |    A346891    |      A039685       |       A346892       |
  |     k = 4    |    A346940    |    100*A067251     |    this sequence    |
  ---------------------------------------------------------------------------
Cf. A346926.

q[n_] := SameQ @@ (d = IntegerDigits[n^2])[[1 ;; 4]] && d[[5]] != d[[1]] && SameQ @@ d[[-4 ;; -1]] && d[[-5]] != d[[-1]]; Select[Range[10000, 3333300], q] (* Amiram Eldar, Aug 08 2021 *)

def ok(n):
  s = str(n*n)
  return len(s) > 4 and s[0] == s[1] == s[2] == s[3] != s[4] and s[-1] == s[-2] == s[-3] == s[-4] != s[-5]
print(list(filter(ok, range(3333333)))) # Michael S. Branicky, Aug 08 2021

A346942_list = [100*n for n in range(99,10**6) if n % 10 and (lambda x:x[0]==x[1]==x[2]==x[3]!=x[4])(str(n**2))] # Chai Wah Wu, Oct 02 2021

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.

A348831 Positive numbers whose square starts and ends with exactly 44, and no 444.

Original entry on oeis.org

212, 2112, 6638, 6662, 6688, 20988, 21012, 21062, 21112, 21138, 21162, 21188, 21212, 66338, 66362, 66388, 66412, 66438, 66488, 66512, 66562, 66588, 66612, 66712, 66738, 66762, 66788, 66812, 66838, 66862, 66888, 66912, 66938, 66988, 67012, 67062, 209762, 209788

Offset: 1

Bernard Schott, Nov 08 2021

When a square starts and ends with digits dd, then dd is necessarily 44.
The last 2 digits of terms are either 12, 38, 62 or 88.
From Marius A. Burtea, Nov 09 2021 : (Start)
The sequence is infinite because the numbers 212, 2112, 21112, ..., (19*10^k + 8) / 9, k >= 3, are terms because the remainder when dividing by 1000 is 544 and 445*10^(2*k - 2) < ((19*10^k + 8) / 9)^2 < 447*10^(2*k - 2), k >= 3.
Also 6638, 66338, 663338, 6633338, 66333338, 663333338, 6633333338, ..., (199*10^k + 14) / 3, k >= 2, are terms and have no digits 0, because their squares are: 44063044, 4400730244, 4400730244, 440017302244, 44001173022244, 4400111730222244, 440011117302222244, ... (End)

			212 is a term since 212^2 = 44944.
662 is not a term since 662^2 = 438244.
668 is not a term since 668^2 = 446224.
2108 is not a term since 2108^2 = 4443664.
21038 is not a term since 21038^2 = 442597444.
21088 is not a term since 21088^2 = 444703744.

Cf. A017317.
Subsequence of A045858, A273375, A305719 and A346774.
Similar to: A348488 (d=4), this sequence (dd=44), A348832 (ddd=444).

fd:=func; fs:=func; [n:n in [1..210000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 08 2021

Select[Range[10, 300000], (d = IntegerDigits[#^2])[[1 ;; 2]] ==  d[[-2 ;; -1]] == {4, 4} && d[[-3]] != 4 && d[[3]] != 4 &] (* Amiram Eldar, Nov 08 2021 *)

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-2
def aupto(N):
  ends = [12, 38, 62, 88]
  r = takewhile(lambda x: x<=N, (100*i+d for i in count(0) for d in ends))
  return [k for k in r if ok(k)]
print(aupto(209788)) # Michael S. Branicky, Nov 08 2021

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

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A346892 Numbers whose square starts and ends with exactly 3 identical digits.

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

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A346942 Numbers whose square starts and ends with exactly 4 identical digits.

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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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A348831 Positive numbers whose square starts and ends with exactly 44, and no 444.

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