A346926 -id:A346926 - cp's OEIS frontend

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

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

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

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