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A379397 Numbers that can be written in exactly two different ways as a sum of at most nine positive third powers.

%I A379397 #23 Dec 23 2024 22:01:41
%S A379397 8,9,16,27,28,29,30,32,33,34,36,37,40,41,42,43,44,48,49,51,54,55,57,
%T A379397 58,59,60,61,62,63,66,69,71,73,74,76,77,78,79,80,85,87,88,90,95,101,
%U A379397 102,103,104,106,109,111,114,115,116,117,122,123,239
%N A379397 Numbers that can be written in exactly two different ways as a sum of at most nine positive third powers.
%C A379397 The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes (cf. A002804).
%H A379397 Wikipedia, <a href="https://en.wikipedia.org/wiki/Waring%27s_problem">Waring's Problem</a>
%H A379397 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>
%H A379397 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%e A379397 29 is in the sequence since 1^3 + 1^3 + 3^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3.
%e A379397 123 is in the sequence since 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 3^3 + 4^3.
%o A379397 (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==2,v,1) \\ _David A. Corneth_, Dec 23 2024
%Y A379397 Cf. A002804, A025453, A379396, A379398, A379399, A379400.
%K A379397 nonn,more
%O A379397 1,1
%A A379397 _Patrick De Geest_, Dec 22 2024