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A379399 Numbers that can be written in exactly four different ways as a sum of at most nine positive third powers.

%I A379399 #23 Dec 29 2024 21:03:25
%S A379399 72,91,126,128,129,131,132,134,135,136,137,138,139,140,144,146,147,
%T A379399 154,155,156,157,158,162,164,165,166,168,170,171,172,173,179,180,181,
%U A379399 184,185,187,191,193,194,195,199,203,205,206,207,210,211,215,221,228,229,230,231,232,241,242,266,267,293,295,319,330,338,366,455
%N A379399 Numbers that can be written in exactly four different ways as a sum of at most nine positive third powers.
%C A379399 The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes (cf. A002804).
%C A379399 Conjecture: this sequence is finite and a(66) = 455 is the last term. Verified up to 10^8. - _Charles R Greathouse IV_, Dec 28 2024
%H A379399 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>.
%H A379399 Wikipedia, <a href="https://en.wikipedia.org/wiki/Waring%27s_problem">Waring's Problem</a>.
%H A379399 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>.
%e A379399 215 is in the sequence since 1^3+2^3+3^3+3^3+3^3+5^3 = 1^3+1^3+2^3+2^3+2^3+4^3+5^3 = 2^3+2^3+3^3+3^3+3^3+3^3+3^3+4^3 = 1^3+2^3+2^3+2^3+2^3+3^3+3^3+4^3+4^3.
%o A379399 (PARI) upto(n) = my(v=vector(n), maxb=sqrtnint(n, 3)); forvec(x=vector(9, i, [0, maxb]), s=sum(i=1, 9, x[i]^3); if(0<s && s<=n, v[s]++); , 1); select(x->x==4, v, 1) \\ _David A. Corneth_, Dec 23 2024
%Y A379399 Cf. A002804, A025453, A379396, A379397, A379398, A379400.
%K A379399 nonn
%O A379399 1,1
%A A379399 _Patrick De Geest_, Dec 23 2024