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.

-1, 0, 3, 0, 9, 15, 0, 35, 39, 118, 0, 33, 31, 463, 90, 0, 17, 138, 558, 200, 435, 0, 63, 57, 1080, 580, 1580, 644, 0, 15, 198, 750, 1375, 2400, 1820, 294, 0, 91, 87, 1200, 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

Offset: 1

Author

Jinyuan Wang, Aug 10 2020

Keywords

• sign
• tabl

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) = 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.

Examples

			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,  ...
 ...
		

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.

Crossrefs

Cf. A000010, A057635, A336385.

Extensions

Terms a(16) onward from Max Alekseyev, Feb 01 2025