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A321292 Smallest positive number for which the 5th power cannot be written as sum of distinct 5th powers of any subset of previous terms.

%I A321292 #13 Sep 19 2024 21:57:03
%S A321292 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,25,26,27,28,30,
%T A321292 37,43,44,55,57,64,77,82,90,97,112,116,119,154,156,178,202,227,269,
%U A321292 309,335,371,397,442,516,604,643,722,774,815,1000,1115,1308,1503
%N A321292 Smallest positive number for which the 5th power cannot be written as sum of distinct 5th powers of any subset of previous terms.
%C A321292 a(n)^5 forms a sum-free sequence.
%H A321292 Bert Dobbelaere, <a href="/A321292/b321292.txt">Table of n, a(n) for n = 1..150</a>
%H A321292 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sum-free_sequence">Sum-free sequence</a>
%e A321292 The smallest number > 0 that is not in the sequence is 12, because
%e A321292 12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5.
%o A321292 (Python)
%o A321292 def findSum(nopt, tgt, a, smax, pwr):
%o A321292     if nopt==0:
%o A321292         return [] if tgt==0 else None
%o A321292     if tgt<0 or tgt>smax[nopt-1]:
%o A321292         return None
%o A321292     rv=findSum(nopt-1, tgt - a[nopt-1]**pwr, a, smax, pwr)
%o A321292     if rv!=None:
%o A321292         rv.append(a[nopt-1])
%o A321292     else:
%o A321292         rv=findSum(nopt-1, tgt, a, smax, pwr)
%o A321292     return rv
%o A321292 def A321292(n):
%o A321292     POWER=5 ; x=0 ; a=[] ; smax=[] ; sumpwr=0
%o A321292     while len(a)<n:
%o A321292         while True:
%o A321292             x+=1
%o A321292             lst=findSum(len(a), x**POWER, a, smax, POWER)
%o A321292             if lst==None:
%o A321292                 break
%o A321292             rhs = " + ".join(["%d^%d"%(i, POWER) for i in lst])
%o A321292             print("    %d^%d = %s"%(x, POWER, rhs))
%o A321292         a.append(x) ; sumpwr+=x**POWER
%o A321292         print("a(%d) = %d"%(len(a), x))
%o A321292         smax.append(sumpwr)
%o A321292     return a[-1]
%Y A321292 Other powers: A321266 (2), A321290 (3), A321291 (4), A321293 (6).
%K A321292 nonn
%O A321292 1,2
%A A321292 _Bert Dobbelaere_, Nov 02 2018