%I #12 Sep 19 2024 21:57:07

%S 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 32,34,36,38,40,42,44,46,48,52,54,56,64,68,72,76,80,84,88,92,96,104,

%U 108,112,128,136,144,152,160,168,176,184,192,208,216,224,256

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

%C a(n)^4 forms a sum-free sequence.

%C 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 Bert Dobbelaere, <a href="/A321291/b321291.txt">Table of n, a(n) for n = 1..104</a>

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

%F a(n) = 2 * a(n-12) for n > 25 (conjectured).

%e The smallest number > 0 that is not in the sequence is 15, because

%e 15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4.

%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 A321291(n):

%o POWER=4 ; 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), A321292 (5), A321293 (6).

%K nonn

%O 1,2

%A _Bert Dobbelaere_, Nov 02 2018