A321266 Smallest positive number for which the square cannot be written as sum of distinct squares of any subset of previous terms.
1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 32, 34, 48, 64, 68, 96, 128, 136, 192, 256, 272, 384, 512, 544, 768, 1024, 1088, 1536, 2048, 2176, 3072, 4096, 4352, 6144, 8192, 8704, 12288, 16384, 17408, 24576, 32768, 34816, 49152, 65536, 69632, 98304, 131072, 139264
Offset: 1
Keywords
Examples
0^2 = 0 (sum of squares of the empty set). 1^2 cannot be written as sum of squares of the empty set, so a(1)=1. Suppose we determined all terms up to a(7)=12: 13^2 = 12^2 + 4^2 + 3^2, 14^2 = 12^2 + 6^2 + 4^2, 15^2 = 12^2 + 8^2 + 4^2 + 1^2. 16^2 cannot be written as sum of squares of distinct smaller terms, hence a(8)=16.
Links
- Bert Dobbelaere, Table of n, a(n) for n = 1..89
- Wikipedia, Sum-free sequence
Programs
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Python
def findSum(nopt, tgt, a, smax, pwr): if nopt==0: return [] if tgt==0 else None if tgt<0 or tgt>smax[nopt-1]: return None rv=findSum(nopt-1, tgt - a[nopt-1]**pwr, a, smax, pwr) if rv!=None: rv.append(a[nopt-1]) else: rv=findSum(nopt-1,tgt, a, smax, pwr) return rv def A321266(n): POWER=2 ; x=0 ; a=[] ; smax=[] ; sumpwr=0 while len(a)
Formula
a(n) = 2 * a(n-3) for n > 9 (conjectured).
Comments