The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A328910

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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.
(history; published version)

#86 by Alois P. Heinz at Sat Jun 26 22:39:27 EDT 2021

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proposed

approved

#85 by Jon E. Schoenfield at Sat Jun 26 20:24:11 EDT 2021

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editing

proposed

#84 by Jon E. Schoenfield at Sat Jun 26 20:24:09 EDT 2021

COMMENTS

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 integer and >= x_3 = 7, we have a new solution: (1, 1, 7, 12). (End)

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*log2log_2(n)). - Max Alekseyev, Nov 10 2019

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approved

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#83 by N. J. A. Sloane at Thu Jun 04 00:48:38 EDT 2020

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approved

#82 by N. J. A. Sloane at Thu Jun 04 00:48:35 EDT 2020

LINKS

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, loc. cit, 127:5 (2020), 478.

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approved

editing

#81 by N. J. A. Sloane at Thu Jun 04 00:45:57 EDT 2020

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approved

#80 by N. J. A. Sloane at Thu Jun 04 00:45:56 EDT 2020

LINKS

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, loc. cit, 127:5 (2020), 478.

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approved

editing

#79 by Max Alekseyev at Sun Nov 10 12:57:28 EST 2019

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editing

approved

#78 by Max Alekseyev at Sun Nov 10 12:57:17 EST 2019

PROG

(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

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approved

editing

#77 by Max Alekseyev at Sun Nov 10 12:35:33 EST 2019

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editing

approved