OFFSET
1,3
COMMENTS
For n = 1, we have phi(x_1) = k*phi(x_1), thus A(1, k) = 0 iff k >= 2.
For n >= 2, if phi(Product_{i=1..n} x_i) = Sum_{i=1..n} k*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.
LINKS
Shi Baohuai and Pan Xiaowei, On the arithmetic functional equation phi(x_1*...*x_(n-1)*x_n) = m*(phi(x_1) + ... + phi(x_(n-1)) + phi(x_n)), Mathematics Practice and Understanding, 2014, Issue 24, Pages 307-310.
EXAMPLE
The square array A(n,k) begins:
-1, 0, 0, 0, 0, ...
3, 9, 35, 33, 17, ...
15, 39, 31, 138, 57, ...
118, 463, 558, 1080, 732, ...
...
CROSSREFS
KEYWORD
AUTHOR
Jinyuan Wang, Aug 10 2020
STATUS
approved