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A191867 Numbers n which are both the sum of two nonzero squares and the concatenation of the decimal representation of two nonzero squares.

%I A191867 #22 Oct 21 2024 01:20:44
%S A191867 41,116,125,136,149,164,169,181,369,416,425,436,449,464,481,641,916,
%T A191867 925,936,949,964,981,1009,1225,1256,1289,1361,1576,1616,1625,1636,
%U A191867 1649,1664,1681,1961,2516,2525,2536,2549,2561,2564,2581,3616,3625,3636,3649,3664
%N A191867 Numbers n which are both the sum of two nonzero squares and the concatenation of the decimal representation of two nonzero squares.
%C A191867 It would be interesting to investigate such numbers w.r.t. higher powers and larger n.
%H A191867 Klaus Brockhaus, <a href="/A191867/b191867.txt">Table of n, a(n) for n = 1..1000</a>
%e A191867 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).
%e A191867 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).
%t A191867 (* find numbers that can be split as the SUM of two powers (squares, cubes, etc.) and also as CONCATENATION of the same powers *)
%t A191867 siamesePowers[n_, power_] := Module[
%t A191867 {listOfSumOfPowers, a, b, i, listOfConcatenatedPowers},
%t A191867 listOfSumOfPowers = Outer[Plus, Table[{i^power}, {i, 1, n}], Table[{i^power}, {i, 1, n}]] // Flatten;
%t A191867 concatNumbers[a_, b_] := IntegerDigits[{a, b}] // Flatten // FromDigits;
%t A191867 listOfConcatenatedPowers := Outer[concatNumbers, Table[i^power, {i, 1, n}], Table[i^power, {i, 1, n}]] // Flatten;
%t A191867 (* The intersection of these 2 lists is the set of our special Siamese numbers *)
%t A191867 Intersection[listOfSumOfPowers, listOfConcatenatedPowers]
%t A191867 ]
%t A191867 siamesePowers[30, 2] (* Generate the first 30 such numbers for squares *)
%o A191867 (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
%o A191867 (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
%Y A191867 Cf. A000404, A048385.
%K A191867 nonn,base
%O A191867 1,1
%A A191867 Raghavendra Ugare, Jun 18 2011