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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.
1
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
1,1
COMMENTS
It would be interesting to investigate such numbers w.r.t. higher powers and larger n.
LINKS
EXAMPLE
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).
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 *)
PROG
(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
CROSSREFS
Sequence in context: A105389 A266954 A290589 * A195317 A044292 A044673
KEYWORD
nonn,base
AUTHOR
Raghavendra Ugare, Jun 18 2011
STATUS
approved