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

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn
• base

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

Examples

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

Links

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

Crossrefs

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

Cf. A055617, A055618, A055619.

Programs

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 *)

PARI

select( {is_A055616(n, L=logint(n,10))=L%2 && n==norml2(divrem(n,10^(L\/2)))}, [1..10^5]) \\ M. F. Hasler, Dec 20 2024
for(L=1,oo, for(n=10^L,10^L++, is_A055616(n)&& print1(n", "))) \\ slow beyond 10^6

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

Extensions

Definition corrected by Derek Orr, Jul 09 2014