A379399 - OEIS

%I #23 Dec 29 2024 21:03:25

%S 72,91,126,128,129,131,132,134,135,136,137,138,139,140,144,146,147,

%T 154,155,156,157,158,162,164,165,166,168,170,171,172,173,179,180,181,

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

%C 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 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Waring%27s_problem">Waring's Problem</a>.

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>.

%e 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 (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 Cf. A002804, A025453, A379396, A379397, A379398, A379400.

%K nonn

%O 1,1

%A _Patrick De Geest_, Dec 23 2024