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A191867
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Numbers n which are both the sum of two nonzero squares and the concatenation of the decimal representation of two nonzero squares.
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1
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41, 116, 125, 136, 149, 164, 169, 181, 369, 416, 425, 436, 449, 464, 481, 641, 916, 925, 936, 949, 964, 981, 1009, 1225, 1256, 1289, 1361, 1576, 1616, 1625, 1636, 1649, 1664, 1681, 1961, 2516, 2525, 2536, 2549, 2561, 2564, 2581, 3616, 3625, 3636, 3649, 3664
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OFFSET
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1,1
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COMMENTS
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It would be interesting to investigate such numbers w.r.t. higher powers and larger n.
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LINKS
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EXAMPLE
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The smallest such number is 41, since it is both the sum of two squares (i.e., 4^2, 5^2} and the concatenation of two squares (i.e., 2^2, 1^2).
3649 also belongs to this sequence because it is sum of two squares (i.e., 60^2, 7^2) and the concatenation of two squares (i.e., 6^2, 7^2).
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MATHEMATICA
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(* find numbers that can be split as the SUM of two powers (squares, cubes, etc.) and also as CONCATENATION of the same powers *)
siamesePowers[n_, power_] := Module[
{listOfSumOfPowers, a, b, i, listOfConcatenatedPowers},
listOfSumOfPowers = Outer[Plus, Table[{i^power}, {i, 1, n}], Table[{i^power}, {i, 1, n}]] // Flatten;
concatNumbers[a_, b_] := IntegerDigits[{a, b}] // Flatten // FromDigits;
listOfConcatenatedPowers := Outer[concatNumbers, Table[i^power, {i, 1, n}], Table[i^power, {i, 1, n}]] // Flatten;
(* The intersection of these 2 lists is the set of our special Siamese numbers *)
Intersection[listOfSumOfPowers, listOfConcatenatedPowers]
]
siamesePowers[30, 2] (* Generate the first 30 such numbers for squares *)
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PROG
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(Magma) z:=65; T:=Sort([ s: a in [b..z], b in [1..z] | s le z^2 where s is a^2+b^2 ]); SplitToSquares:=function(n); V:=[]; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then Append(~V, [<b, a>]); end if; end for; return V; end function; U:=[ p: j in [1..#T] | P ne [] where P is SplitToSquares(p) where p is T[j] ]; [ U[j]: j in [1..#U] | j eq 1 or U[j-1] ne U[j] ]; // Klaus Brockhaus, Jun 19 2011
(PARI) is_A191867(n) = for(p=10, n, issquare(n\p) && issquare(n%p) && n%p*10>=p && return(is_A000404(n)); p=p*10-1) \\ M. F. Hasler, Jun 19 2011
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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Raghavendra Ugare, Jun 18 2011
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STATUS
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approved
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