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%I #17 Oct 05 2021 00:30:25
%S 2,6,9,11,12,15,16,18,20,21,24,29,32,33,36,41,42,45,48,50,51,56,57,60,
%T 66,72,76,77,81,82,84,90,96,99,101,102,105,106,108,113,114,120,122,
%U 123,126,132,136,137,140,141,144,146,156,162,164,168,171,176,177,180
%N Numbers k such that there is no nonzero integer solution for the Diophantine equation x_1^2 + x_2^2 + ... + x_k^2 = x_1*x_2*...*x_k.
%C In the link, a C++ program calling the GMP library is provided to solve such kinds of equations.
%C If the equation has nonzero solutions and k > 2, then there is a positive integer solution (x_1, x_2, ..., x_k) such that 3 <= x_1*x_2*...*x_(k-2) <= n and x_(k-1) <= sqrt((x_1^2 + x_2^2 + ... + x_(k-2)^2)/(x_1*x_2*...*x_(k-2) - 2)).
%H <a href="http://zdu.spaces.live.com/blog/cns!C95152CB25EF2037!164.entry">Link for the problem to be solved</a> [broken link?]
%H <a href="http://topic.csdn.net/u/20081102/00/019ea61c-8ef6-4dd3-a66d-971e4ff1e0ea.html">A Chinese webpage where the problem is raised</a>
%e For k=3, there are nonzero integer solutions 3^2 + 3^2 + 3^2 = 3*3*3; 3^2 + 6^2 + 15^2 = 3*6*15.
%e For k=4, there are nonzero integer solutions 2^2 + 2^2 + 2^2 + 2^2 = 2*2*2*2; 2^2 + 6^2 + 22^2 + 262^2 = 2*6*22*262.
%e However, for k=2, there is no nonzero integer solution for the equation a^2 + b^2 = a*b.
%o (PARI) is(w, k) = my(p, s); for(x=w[k], sqrtint((s=sum(i=1, k, w[i]^2))\p=vecprod(w)-2), if(issquare((p^2+4*p)*x^2-4*s), return(1)))
%o lista(nn) = my(b, t, v=List([])); for(n=2, nn, b=1; for(i=1, #v, if(n%vecprod(v[i])==0&&v[i][1]<=t=n\vecprod(v[i]), listput(v, concat(t, v[i])))); listput(v, [n]); for(m=2, #v, if(is(concat(vector(n-2-#v[m], i, 1), v[m]), n-2), b=0; break)); if(b, print1(n, ", "))) \\ _Jinyuan Wang_, Oct 04 2021
%Y Cf. A002559, A061292.
%K nonn
%O 1,1
%A _Zhao Hui Du_, Nov 04 2008
%E Edited by _Jon E. Schoenfield_, Aug 09 2015
%E More terms from _Jinyuan Wang_, Oct 04 2021
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