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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. 5

%I

%S 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 29,30,31,33,34,42,43,51,57,60,61,71,74,88,91,99,112,116,117,132,152,

%U 153,176,203,228,244,256,281,293,345,392,439,441,529,594,627

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

%C a(n)^6 forms a sum-free sequence.

%H Bert Dobbelaere, <a href="/A321293/b321293.txt">Table of n, a(n) for n = 1..150</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sum-free_sequence">Sum-free sequence</a>

%e 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 (Python)

%o def findSum(nopt, tgt, a, smax, pwr):

%o ....if nopt==0:

%o ........return [] if tgt==0 else None

%o ....if tgt<0 or tgt>smax[nopt-1]:

%o ........return None

%o ....rv=findSum(nopt-1, tgt - a[nopt-1]**pwr, a, smax, pwr)

%o ....if rv!=None:

%o ........rv.append(a[nopt-1])

%o ....else:

%o ........rv=findSum(nopt-1,tgt, a, smax, pwr)

%o ....return rv

%o def A321293(n):

%o ....POWER=6 ; x=0 ; a=[] ; smax=[] ; sumpwr=0

%o ....while len(a)<n:

%o ........while True:

%o ............x+=1

%o ............lst=findSum(len(a), x**POWER, a, smax, POWER)

%o ............if lst==None:

%o ................break

%o ............rhs = " + ".join(["%d^%d"%(i,POWER) for i in lst])

%o ............print(" %d^%d = %s"%(x,POWER,rhs))

%o ........a.append(x) ; sumpwr+=x**POWER

%o ........print("a(%d) = %d"%(len(a),x))

%o ........smax.append(sumpwr)

%o ....return a[-1]

%Y Other powers: A321266 (2), A321290 (3), A321291 (4), A321292 (5).

%K nonn

%O 1,2

%A _Bert Dobbelaere_, Nov 02 2018

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Last modified June 16 15:18 EDT 2021. Contains 345057 sequences. (Running on oeis4.)