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A321293 Smallest positive number for which the 6th power cannot be written as sum of distinct 6th powers of any subset of previous terms.

%I A321293 #11 Sep 19 2024 21:56:59
%S A321293 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,26,27,
%T A321293 29,30,31,33,34,42,43,51,57,60,61,71,74,88,91,99,112,116,117,132,152,
%U A321293 153,176,203,228,244,256,281,293,345,392,439,441,529,594,627
%N A321293 Smallest positive number for which the 6th power cannot be written as sum of distinct 6th powers of any subset of previous terms.
%C A321293 a(n)^6 forms a sum-free sequence.
%H A321293 Bert Dobbelaere, <a href="/A321293/b321293.txt">Table of n, a(n) for n = 1..150</a>
%H A321293 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sum-free_sequence">Sum-free sequence</a>
%e A321293 The smallest number > 0 that is not in the sequence is 25, because 25^6 = 1^6 + 2^6 + 3^6 + 5^6 + 6^6 + 7^6 + 8^6 + 9^6 + 10^6 + 12^6 + 13^6 + 15^6 + 16^6 + 17^6 + 18^6 + 23^6.
%o A321293 (Python)
%o A321293 def findSum(nopt, tgt, a, smax, pwr):
%o A321293     if nopt==0:
%o A321293         return [] if tgt==0 else None
%o A321293     if tgt<0 or tgt>smax[nopt-1]:
%o A321293         return None
%o A321293     rv=findSum(nopt-1, tgt - a[nopt-1]**pwr, a, smax, pwr)
%o A321293     if rv!=None:
%o A321293         rv.append(a[nopt-1])
%o A321293     else:
%o A321293         rv=findSum(nopt-1, tgt, a, smax, pwr)
%o A321293     return rv
%o A321293 def A321293(n):
%o A321293     POWER=6 ; x=0 ; a=[] ; smax=[] ; sumpwr=0
%o A321293     while len(a)<n:
%o A321293         while True:
%o A321293             x+=1
%o A321293             lst=findSum(len(a), x**POWER, a, smax, POWER)
%o A321293             if lst==None:
%o A321293                 break
%o A321293             rhs = " + ".join(["%d^%d"%(i, POWER) for i in lst])
%o A321293             print("    %d^%d = %s"%(x, POWER, rhs))
%o A321293         a.append(x) ; sumpwr+=x**POWER
%o A321293         print("a(%d) = %d"%(len(a), x))
%o A321293         smax.append(sumpwr)
%o A321293     return a[-1]
%Y A321293 Other powers: A321266 (2), A321290 (3), A321291 (4), A321292 (5).
%K A321293 nonn
%O A321293 1,2
%A A321293 _Bert Dobbelaere_, Nov 02 2018