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

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

Offset: 1

Author

Raghavendra Ugare, Jun 18 2011

Keywords

• nonn
• base

Comments

It would be interesting to investigate such numbers w.r.t. higher powers and larger n.

Examples

			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).
		

Links

• Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000404, A048385.

Programs

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, []); 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

Mathematica

(* 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 *)

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