cp's OEIS Frontend

A351199 Least positive integer m such that m^3*n = x^3 + y^3 + z^3 for some nonnegative integers x,y,z.

%I A351199 #30 Mar 07 2023 18:51:30
%S A351199 1,1,1,1,15,6,5,3,1,1,1,11,39,3,3,3,1,1,4,2,2,3,18,6,1,22,28,1,1,1,29,
%T A351199 15,15,21,3,1,1,7,7,25,3,12,6,1,2,7,2,7,5,21,6,2,25,5,1,1,3,3,45,132,
%U A351199 6,45,1,3,1,1,1,171,6,9,2,3,1,1,54,21,18,3,13,32,1,1,10,2,7,9,3,3,6,3,11,1,1,63,3,30,21,5,4,1,12
%N A351199 Least positive integer m such that m^3*n = x^3 + y^3 + z^3 for some nonnegative integers x,y,z.
%C A351199 a(n) always exists, because any positive rational number can be written as a sum of three cubes of positive rational numbers (see Richmond reference).
%D A351199 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th Edition, Oxford Univ. Press, 1960. (See Theorem 234 on page 197.)
%H A351199 Charles R Greathouse IV, <a href="/A351199/b351199.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..600 from Zhi-Wei Sun)
%H A351199 H. W. Richmond, <a href="https://doi.org/10.1112/plms/s2-21.1.401">On analogues of Waring's problem for rational numbers</a>, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 401-409.
%H A351199 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
%H A351199 Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four rational squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020-2022.
%F A351199 a(n) >= sqrt(A351179(n)).
%e A351199 a(4) = 15 with 15^3*4 = 12^3 + 17^3 + 19^3.
%e A351199 a(212) = 216 with 216^3*212 = 82^3 + 161^3 + 1287^3.
%e A351199 a(446) = 228 with 228^3*446 = 929^3 + 1287^3 + 1330^3.
%t A351199 CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
%t A351199 tab={};Do[m=1; Label[bb]; k=m^3; Do[If[CQ[k*n-x^3-y^3], tab=Append[tab,m]; Goto[aa]],  {x, 0, ((k*n)/3)^(1/3)}, {y, x, ((k*n-x^3)/2)^(1/3)}];
%t A351199 m=m+1; Goto[bb]; Label[aa], {n, 0, 100}]
%o A351199 (PARI) T=thueinit('x^3+1);
%o A351199 has2(n)=n==0 || #select(v->min(v[1], v[2])>=0, thue(T, n))>0
%o A351199 has3(n)=forstep(k=sqrtnint(n,3),sqrtnint(n\3,3),-1,if(has2(n-k^3),return(1)));0
%o A351199 a(n)=my(m=1); while(!has3(m^3*n), m++); m \\ _Charles R Greathouse IV_, Feb 05 2022
%Y A351199 Cf. A000578, A350714, A351179.
%K A351199 nonn
%O A351199 0,5
%A A351199 _Zhi-Wei Sun_, Feb 04 2022