A055616 Numbers, with an even number of digits, that are the sum of the squares of their two halves (leading zeros allowed only for the second half).

1233, 8833, 990100, 94122353, 1765038125, 2584043776, 7416043776, 8235038125, 9901009901, 116788321168, 123288328768, 876712328768, 883212321168, 999900010000, 13793103448276, 15348303604525, 84651703604525, 86206903448276, 91103202846976, 92318202663025

Offset

1,1

Comments

The sequence is infinite since it contains several infinite subsequences (see A055617, etc.).

If x = A*10^n+B is an element not beginning with 9, then (10^n-A)*10^n+B is another (e.g. 1233 <-> 8833).

Numbers that can be written as n = A*10^d + B with 10^(d-1) <= A < 10^d, 0 <= B < 10^d, and A^2 + B^2 = n. - Robert Israel, May 10 2015

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

8833 is ok, since 8833 = 88^2 + 33^2.

Maple

dmax:= 8: # to get all entries with at most 2*dmax digits
Res:= NULL:
for d from 2 to dmax  do
     cands:= map(t -> subs(t, [x, y]), [isolve(x^2 + y^2 = 10^(2*d)+1)]);
     cands:= select(t -> t[1]::even and t[1]>=0 and t[2]>0, cands);
     cands:= map(t -> ([(10^d + t[1])/2, (t[2]+1)/2], [(10^d-t[1])/2, (t[2]+1)/2]), cands);
     cands:= select(t -> (t[1]>= 10^(d-1) and t[1] < 10^d and t[2] <= 10^d), cands);
     Res:= Res, op(map(t -> 10^d*t[1]+t[2], cands));
od:
sort([Res]); # Robert Israel, May 10 2015

Mathematica

fQ[n_] := Block[{d = IntegerDigits@ n}, If[OddQ[Length@ d], False, Plus[FromDigits[Take[d, Length[d]/2]]^2, FromDigits[Take[d, -Length[d]/2]]^2]] == n]; Select[Range@ 1000000, fQ] (* Michael De Vlieger, May 09 2015 *)

Prog

(Python)
def a():
..n = 1
..while n < 10**6:
....st = str(n)
....if len(st) % 2 == 0:
......s1 = st[:int(len(st)/2)]
......s2 = st[int(len(st)/2):int(len(st))]
......if int(s1)**2+int(s2)**2 == int(st):
........print(n, end=', ')
........n += 1
......else:
........n += 1
....else:
......n = 10*n
a()
# Derek Orr, Jul 08 2014

Crossrefs

Cf. A064942 for the number of solutions, where leading zeros are allowed.

Cf. A055617, A055618, A055619.

Sequence in context: A206272 A064942 A101311 * A104971 A193492 A279204

Adjacent sequences: A055613 A055614 A055615 * A055617 A055618 A055619

Keyword

nonn,base

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Extensions

Definition corrected by Derek Orr, Jul 09 2014

Status

approved