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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
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)).
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][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
CROSSREFS
Sequence in context: A071814 A121248 A066586 * A133160 A128906 A192420
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