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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
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: A370810 A071814 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

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Last modified March 29 08:53 EDT 2024. Contains 371268 sequences. (Running on oeis4.)