cp's OEIS Frontend

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).

%I A055616 #32 Dec 20 2024 10:06:51
%S A055616 1233,8833,990100,94122353,1765038125,2584043776,7416043776,
%T A055616 8235038125,9901009901,116788321168,123288328768,876712328768,
%U A055616 883212321168,999900010000,13793103448276,15348303604525,84651703604525,86206903448276,91103202846976,92318202663025
%N 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).
%C A055616 The sequence is infinite since it contains several infinite subsequences (see A055617, etc.).
%C A055616 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).
%C A055616 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
%H A055616 Robert Israel, <a href="/A055616/b055616.txt">Table of n, a(n) for n = 1..1000</a>
%e A055616 8833 is ok, since 8833 = 88^2 + 33^2.
%p A055616 dmax:= 8: # to get all entries with at most 2*dmax digits
%p A055616 Res:= NULL:
%p A055616 for d from 2 to dmax  do
%p A055616      cands:= map(t -> subs(t,[x,y]), [isolve(x^2 + y^2 = 10^(2*d)+1)]);
%p A055616      cands:= select(t -> t[1]::even and t[1]>=0 and t[2]>0, cands);
%p A055616      cands:= map(t -> ([(10^d + t[1])/2, (t[2]+1)/2], [(10^d-t[1])/2, (t[2]+1)/2]), cands);
%p A055616      cands:= select(t -> (t[1]>= 10^(d-1) and t[1] < 10^d and t[2] <= 10^d), cands);
%p A055616      Res:= Res, op(map(t -> 10^d*t[1]+t[2], cands));
%p A055616 od:
%p A055616 sort([Res]); # _Robert Israel_, May 10 2015
%t A055616 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 *)
%o A055616 (Python)
%o A055616 def a():
%o A055616   n = 1
%o A055616   while n < 10**6:
%o A055616     st = str(n)
%o A055616     if len(st) % 2 == 0:
%o A055616       s1 = st[:int(len(st)/2)]
%o A055616       s2 = st[int(len(st)/2):int(len(st))]
%o A055616       if int(s1)**2+int(s2)**2 == int(st):
%o A055616         print(n,end=', ')
%o A055616         n += 1
%o A055616       else:
%o A055616         n += 1
%o A055616     else:
%o A055616       n = 10*n
%o A055616 a()
%o A055616 # _Derek Orr_, Jul 08 2014
%o A055616 (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
%o A055616 for(L=1,oo, for(n=10^L,10^L++, is_A055616(n)&& print1(n", "))) \\ slow beyond 10^6
%Y A055616 Cf. A064942 for the number of solutions, where leading zeros are allowed.
%Y A055616 Cf. A055617, A055618, A055619.
%K A055616 nonn,base
%O A055616 1,1
%A A055616 Ulrich Schimke (ulrschimke(AT)aol.com)
%E A055616 Definition corrected by _Derek Orr_, Jul 09 2014