A336710 - OEIS

A336710

Square array read by antidiagonals: A(n,k) is the number of ordered solutions (x_1, x_2, ..., x_n) to equation phi(Product_{i=1..n} x_i) = k * Sum_{i=1..n} phi(x_i), or -1 if there are infinitely many solutions, n >= 1, k >= 1.

%I #18 Feb 02 2025 04:29:20

%S -1,0,3,0,9,15,0,35,39,118,0,33,31,463,90,0,17,138,558,200,435,0,63,

%T 57,1080,580,1580,644,0,15,198,750,1375,2400,1820,294,0,91,87,1200,

%U 570,4695,3535,3024,792,0,79,411,528,2490,1680,8386,12292,5256,3285,0,67,183,2584,685,7710,2555,15568,14364,16605,1595,0,39,294,1346,6565,2790,21070,6160,42030,28305,21780,15708,0

%N Square array read by antidiagonals: A(n,k) is the number of ordered solutions (x_1, x_2, ..., x_n) to equation phi(Product_{i=1..n} x_i) = k * Sum_{i=1..n} phi(x_i), or -1 if there are infinitely many solutions, n >= 1, k >= 1.

%C For n = 1, we have phi(x_1) = k * phi(x_1), thus A(1, k) = 0 iff k >= 2.

%C For n >= 2, if phi(Product_{i=1..n} x_i) = k * Sum_{i=1..n} phi(x_i) and phi(x_1) <= phi(x_2) <= ... <= phi(x_n), then phi(x_(n-1)) <= n*k and phi(x_n) <= k*(n-1)*phi(x_(n-1)). It implies that the equation has finite solutions iff n >= 2 or k >= 2.

%H Shi Baohuai and Pan Xiaowei, <a href="http://www.cqvip.com/QK/93074X/201424/663357483.html">On the arithmetic functional equation phi(x_1*...*x_(n-1)*x_n) = m*(phi(x_1) + ... + phi(x_(n-1)) + phi(x_n))</a>, Mathematics Practice and Understanding, 2014, Issue 24, Pages 307-310.

%e The square array A(n,k) begins:

%e -1, 0, 0, 0, 0, ...

%e 3, 9, 35, 33, 17, ...

%e 15, 39, 31, 138, 57, ...

%e 118, 463, 558, 1080, 732, ...

%e ...

%Y Cf. A000010, A057635, A336385.

%K sign,tabl

%O 1,3

%A _Jinyuan Wang_, Aug 10 2020

%E Terms a(16) onward from _Max Alekseyev_, Feb 01 2025