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A146974
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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.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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)).
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LINKS
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Table of n, a(n) for n=1..60.
Link for the problem to be solved [broken link?]
A Chinese webpage where the problem is raised
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EXAMPLE
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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.
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PROG
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(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][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
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CROSSREFS
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Cf. A002559, A061292.
Sequence in context: A109600 A071814 A066586 * A133160 A128906 A192420
Adjacent sequences: A146971 A146972 A146973 * A146975 A146976 A146977
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KEYWORD
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nonn
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AUTHOR
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Zhao Hui Du, Nov 04 2008
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EXTENSIONS
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Edited by Jon E. Schoenfield, Aug 09 2015
More terms from Jinyuan Wang, Oct 04 2021
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STATUS
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approved
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