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A321291 Smallest positive number for which the 4th power cannot be written as sum of 4th powers of any subset of previous terms.

%I A321291 #12 Sep 19 2024 21:57:07
%S A321291 1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,21,22,23,24,26,27,28,
%T A321291 32,34,36,38,40,42,44,46,48,52,54,56,64,68,72,76,80,84,88,92,96,104,
%U A321291 108,112,128,136,144,152,160,168,176,184,192,208,216,224,256
%N A321291 Smallest positive number for which the 4th power cannot be written as sum of 4th powers of any subset of previous terms.
%C A321291 a(n)^4 forms a sum-free sequence.
%C A321291 It is noteworthy that the terms of this sequence increase slower than those of similar sequences for smaller (A321266, A321290) but also larger powers (A321292, A321293).
%H A321291 Bert Dobbelaere, <a href="/A321291/b321291.txt">Table of n, a(n) for n = 1..104</a>
%H A321291 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sum-free_sequence">Sum-free sequence</a>
%F A321291 a(n) = 2 * a(n-12) for n > 25 (conjectured).
%e A321291 The smallest number > 0 that is not in the sequence is 15, because
%e A321291     15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4.
%o A321291 (Python)
%o A321291 def findSum(nopt, tgt, a, smax, pwr):
%o A321291     if nopt==0:
%o A321291         return [] if tgt==0 else None
%o A321291     if tgt<0 or tgt>smax[nopt-1]:
%o A321291         return None
%o A321291     rv=findSum(nopt-1, tgt - a[nopt-1]**pwr, a, smax, pwr)
%o A321291     if rv!=None:
%o A321291         rv.append(a[nopt-1])
%o A321291     else:
%o A321291         rv=findSum(nopt-1, tgt, a, smax, pwr)
%o A321291     return rv
%o A321291 def A321291(n):
%o A321291     POWER=4 ; x=0 ; a=[] ; smax=[] ; sumpwr=0
%o A321291     while len(a)<n:
%o A321291         while True:
%o A321291             x+=1
%o A321291             lst=findSum(len(a), x**POWER, a, smax, POWER)
%o A321291             if lst==None:
%o A321291                 break
%o A321291             rhs = " + ".join(["%d^%d"%(i, POWER) for i in lst])
%o A321291             print("    %d^%d = %s"%(x, POWER, rhs))
%o A321291         a.append(x) ; sumpwr+=x**POWER
%o A321291         print("a(%d) = %d"%(len(a), x))
%o A321291         smax.append(sumpwr)
%o A321291     return a[-1]
%Y A321291 Other powers: A321266 (2), A321290 (3), A321292 (5), A321293 (6).
%K A321291 nonn
%O A321291 1,2
%A A321291 _Bert Dobbelaere_, Nov 02 2018