A336710 Square array read by diagonals: A(n,k) is the number of solutions (x_1, x_2, ..., x_n) to equation phi(Product_{i=1..n} x_i) = Sum_{i=1..n} k*phi(x_i), or -1 if there are infinitely many solutions.

-1, 0, 3, 0, 9, 15, 0, 35, 39, 118, 0, 33, 31, 463, 90

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

Table of n, a(n) for n=1..15.

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

Cf. A000010, A057635, A336385.

Sequence in context: A187427 A167352 A318303 * A294106 A094472 A347999

Adjacent sequences: A336707 A336708 A336709 * A336711 A336712 A336713

Keyword

sign,tabl,more

Author

Jinyuan Wang, Aug 10 2020

Status

approved