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A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1*...*x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

%I A328910 #90 Jan 11 2025 13:47:16
%S A328910 2,6,8,8,17,14,19,27,25,15,33,16,30,43,45,18,55,24,43,55,43,22,92,43,
%T A328910 35,68,69,25,107,34,80,56,48,61,130,32,45,65,119,29,113,36,72,154,64,
%U A328910 27,178,69,87,74,85,24,197,97,145,91,51,37,182,39,54,189,203,82,173
%N A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1*...*x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.
%C A328910 For n = 1, any (x_1) would be a solution, therefore the offset is 2.
%C A328910 For any n, the equation would also admit the trivial zero solution (0,...,0). If any x_k = 0, then all x_i must be zero, so there is no other solution in the nonnegative integers.
%C A328910 This sequence gives row sums of A328911 which lists the number of solutions with given number of components > 1.
%C A328910 The Shiu paper gives different values for f(2), f(3) and f(8) than those given here, but the author has confirmed (personal communication) that 2, 6 and 19 are correct.
%C A328910 The multiset [1^(n-2),n+1,n*(2*n-1)] is the solution with maximum product. - _Hugo Pfoertner_, Nov 08 2019
%C A328910 From _David A. Corneth_, Nov 08 2019: (Start)
%C A328910 Given x_1,...,x_{n-1}, one can find the value x_n by solving the resulting linear univariate equation.
%C A328910 For example, for n = 4, if we are given (1, 1, 7, x_4) then we can solve 4*(9 + x_4) = 7*x_4, getting 36 = 3*x_4, i.e., x_4 = 12. As 12 is an integer and >= x_3 = 7, we have a new solution: (1, 1, 7, 12). (End)
%C A328910 In any solution, we have x_1*...*x_{n-1} <= n^2, implying that a(n) is finite for all n > 1. Furthermore, x_1 = x_2 = ... = x_k = 1 for k = n - 1 - floor(2*log_2(n)). - _Max Alekseyev_, Nov 10 2019
%D A328910 Lars Blomberg, Posting to the Sequence Fans Mailing List, Nov 07 2019, seems to have been the first person to notice that there were problems with the published values (given in A328980). - _N. J. A. Sloane_, Nov 08 2019
%D A328910 R. K. Guy, Sum equals product. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter D24, (2004), 299-301 (citing Erdős's question dated Aug 19, 1996)
%H A328910 David A. Corneth, <a href="/A328910/b328910.txt">Table of n, a(n) for n = 2..6250</a>
%H A328910 David A. Corneth, <a href="/A328910/a328910_1.gp.txt">List of solutions and product of variables of solutions for n = 2..160, omitting ones</a>
%H A328910 Peter Shiu, <a href="https://doi.org/10.1080/00029890.2019.1639466">On Erdős's Last Equation</a>, Amer. Math. Monthly, 126 (2019), 802-808; correction, 127:5 (2020), 478.
%e A328910 For n = 2 variables, we have the equation x1*x2 = 2*(x1 + x2) with positive integer solutions (3,6) and (4,4).
%e A328910 For n = 3, we have 3 solutions in C_1(3) = {(1, 4, 15), (1, 5, 9), (1, 6, 7)} (with 2 components > 1), and 3 others in C_2(3) = {(2, 2, 12), (2, 3, 5), (3,3,3)} (with 3 components > 1), for a total of a(3) = 6.
%e A328910 For n = 4 we have the 8 solutions (1, 1, 5, 28), (1, 1, 6, 16), (1, 1, 7, 12), (1, 1, 8, 10), (1, 2, 3, 12), (1, 2, 4, 7), (1, 3, 4, 4) and (2, 2, 2, 6).
%e A328910 For n = 5, the solutions are, omitting initial components x_i = 1: {(6, 45), (7, 25), (9, 15), (10, 13), (2, 3, 35), (2, 5, 9), (3, 3, 10), (3, 5, 5)}.
%e A328910 For n = 6, the solutions are (omitting x_i = 1): {(7, 66), (8, 36), (9, 26), (10, 21), (11, 18), (12, 16), (2, 4, 27), (2, 5, 15), (2, 6, 11), (2, 7, 9), (3, 3, 18), (3, 4, 10), (3, 6, 6), (2, 2, 2, 24), (2, 2, 3, 9), (2, 2, 4, 6), (2, 3, 3, 5)}.
%e A328910 For n = 9, the 27 solutions are (omitting '1's): {(10, 153), (11, 81), (12, 57), (13, 45), (15, 33), (17, 27), (18, 25), (21, 21), (2, 5, 117), (2, 6, 42), (2, 7, 27), (2, 9, 17), (2, 12, 12), (3, 4, 39), (3, 5, 21), (3, 6, 15), (3, 7, 12), (3, 9, 9), (4, 4, 18), (5, 5, 9), (6, 6, 6), (2, 2, 3, 36), (2, 2, 6, 9), (2, 3, 3, 13), (3, 3, 3, 7), (3, 3, 4, 5), (2, 3, 3, 3, 3)}.
%o A328910 (PARI) a(n,show=1)={my(s=0,d);forvec(x=vector(n-1,i,[1,n\(sqrt(2)-1)]), 0<(d=vecprod(x)-n) && n*vecsum(x)%d==0 && n*vecsum(x)\d >= x[n-1] &&s++ &&show &&printf("%d,",concat(x,n*vecsum(x)\d)),1);s}
%o A328910 (PARI) { A328910(n,k=n-1,m=n^2,p=1,s=0,y=1) = if(k==0, return( p>n && Mod(n*s,p-n)==0 && n*s>=(p-n)*y ) ); sum(x=y, sqrtnint(m,k), A328910(n,k-1,m\x,p*x,s+x,x) ); } \\ _Max Alekseyev_, Nov 10 2019
%Y A328910 Row sums of A328911.
%Y A328910 Cf. A000384, A002414, A213820 (apparently maximum product occurring in solutions), A328980, A329205, A329206.
%K A328910 nonn
%O A328910 2,1
%A A328910 _M. F. Hasler_, Nov 07 2019
%E A328910 More terms from _David A. Corneth_, Nov 07 2019