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A174499 Smallest number whose square starts and ends with n identical digits. 0
1, 88, 10538, 235700, 10541000, 57735000, 7453560000, 14907120000, 18257418600000, 235702260400000, 298142397000000, 7453559925000000, 14907119850000000, 105409255338950000000, 7453559924999300000000, 10540925533894600000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For n > 3 the n last identical digits equals zero. 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 of 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).

LINKS

Table of n, a(n) for n=1..16.

FORMULA

For n > 3, a(n) = A119511(n)*10^q, q = floor(n+1)/2.

EXAMPLE

a(3) = 10538 because 10538^2 = 111049444 starts and ends in 3 identical digits.

MAPLE

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:

CROSSREFS

Cf. A119511, A186438, A186439.

Sequence in context: A188991 A189201 A052069 * A048919 A159718 A157460

Adjacent sequences:  A174496 A174497 A174498 * A174500 A174501 A174502

KEYWORD

nonn,base

AUTHOR

Michel Lagneau, Feb 22 2011

STATUS

approved

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Last modified November 18 09:55 EST 2019. Contains 329261 sequences. (Running on oeis4.)