cp's OEIS Frontend

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 A336710 #18 Feb 02 2025 04:29:20
%S A336710 -1,0,3,0,9,15,0,35,39,118,0,33,31,463,90,0,17,138,558,200,435,0,63,
%T A336710 57,1080,580,1580,644,0,15,198,750,1375,2400,1820,294,0,91,87,1200,
%U A336710 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 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.
%C A336710 For n = 1, we have phi(x_1) = k * phi(x_1), thus A(1, k) = 0 iff k >= 2.
%C A336710 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 A336710 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 A336710 The square array A(n,k) begins:
%e A336710   -1,   0,   0,    0,   0,  ...
%e A336710    3,   9,  35,   33,  17,  ...
%e A336710   15,  39,  31,  138,  57,  ...
%e A336710  118, 463, 558, 1080, 732,  ...
%e A336710  ...
%Y A336710 Cf. A000010, A057635, A336385.
%K A336710 sign,tabl
%O A336710 1,3
%A A336710 _Jinyuan Wang_, Aug 10 2020
%E A336710 Terms a(16) onward from _Max Alekseyev_, Feb 01 2025