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 A146974 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. 0
 2, 6, 9, 11, 12, 15, 16, 18, 20, 21, 24, 29, 32, 33, 36, 41, 42, 45, 48, 50, 51, 56, 57, 60, 66, 72, 76, 77, 81, 82, 84, 90, 96, 99, 101, 102, 105, 106, 108, 113, 114, 120, 122, 123, 126, 132, 136, 137, 140, 141, 144, 146, 156, 162, 164, 168, 171, 176, 177, 180 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS In the link, a C++ program calling the GMP library is provided to solve such kinds of equations. 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)). LINKS Link for the problem to be solved [broken link?] EXAMPLE 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. 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. However, for k=2, there is no nonzero integer solution for the equation a^2 + b^2 = a*b. PROG (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))) 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]<=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 CROSSREFS Cf. A002559, A061292. Sequence in context: A109600 A071814 A066586 * A133160 A128906 A192420 Adjacent sequences: A146971 A146972 A146973 * A146975 A146976 A146977 KEYWORD nonn AUTHOR Zhao Hui Du, Nov 04 2008 EXTENSIONS Edited by Jon E. Schoenfield, Aug 09 2015 More terms from Jinyuan Wang, Oct 04 2021 STATUS approved

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Last modified March 30 04:44 EDT 2023. Contains 361603 sequences. (Running on oeis4.)