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A321266 Smallest positive number for which the square cannot be written as sum of distinct squares of any subset of previous terms.

%I A321266 #15 May 22 2025 10:21:48
%S A321266 1,2,3,4,6,8,12,16,17,24,32,34,48,64,68,96,128,136,192,256,272,384,
%T A321266 512,544,768,1024,1088,1536,2048,2176,3072,4096,4352,6144,8192,8704,
%U A321266 12288,16384,17408,24576,32768,34816,49152,65536,69632,98304,131072,139264
%N A321266 Smallest positive number for which the square cannot be written as sum of distinct squares of any subset of previous terms.
%C A321266 a(n)^2 = A226076(n) forms a sum-free sequence.
%H A321266 Bert Dobbelaere, <a href="/A321266/b321266.txt">Table of n, a(n) for n = 1..89</a>
%H A321266 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sum-free_sequence">Sum-free sequence</a>
%F A321266 a(n) = 2 * a(n-3) for n > 9 (conjectured).
%e A321266 0^2 = 0 (sum of squares of the empty set).
%e A321266 1^2 cannot be written as sum of squares of the empty set, so a(1)=1.
%e A321266 Suppose we determined all terms up to a(7)=12:
%e A321266 13^2 = 12^2 + 4^2 + 3^2,
%e A321266 14^2 = 12^2 + 6^2 + 4^2,
%e A321266 15^2 = 12^2 + 8^2 + 4^2 + 1^2.
%e A321266 16^2 cannot be written as sum of squares of distinct smaller terms, hence a(8)=16.
%o A321266 (Python)
%o A321266 def findSum(nopt, tgt, a, smax, pwr):
%o A321266     if nopt==0:
%o A321266         return [] if tgt==0 else None
%o A321266     if tgt<0 or tgt>smax[nopt-1]:
%o A321266         return None
%o A321266     rv=findSum(nopt-1, tgt - a[nopt-1]**pwr, a, smax, pwr)
%o A321266     if rv!=None:
%o A321266         rv.append(a[nopt-1])
%o A321266     else:
%o A321266         rv=findSum(nopt-1,tgt, a, smax, pwr)
%o A321266     return rv
%o A321266 def A321266(n):
%o A321266     POWER=2 ; x=0 ; a=[] ; smax=[] ; sumpwr=0
%o A321266     while len(a)<n:
%o A321266         while True:
%o A321266             x+=1
%o A321266             lst=findSum(len(a), x**POWER, a, smax, POWER)
%o A321266             if lst==None:
%o A321266                 break
%o A321266             rhs = " + ".join(["%d^%d"%(i,POWER) for i in lst])
%o A321266             print("    %d^%d = %s"%(x,POWER,rhs))
%o A321266         a.append(x) ; sumpwr+=x**POWER
%o A321266         print("a(%d) = %d"%(len(a),x))
%o A321266         smax.append(sumpwr)
%o A321266     return a[-1]
%Y A321266 Square root of A226076.
%Y A321266 Other powers: A321290 (3), A321291 (4), A321292 (5), A321293 (6).
%K A321266 nonn
%O A321266 1,2
%A A321266 _Bert Dobbelaere_, Nov 01 2018