MathDB

The

n

-tuple

(a_1,a_2,\ldots,a_n)

of integers satisfies the following: (i)

1\le a_1<a_2<\cdots < a_n\le 50

(ii) for each

n

-tuple

(b_1,b_2,\ldots,b_n)

of positive integers, there exist a positive integer

m

and an

n

-tuple

(c_1,c_2,\ldots,c_n)

of positive integers such that

mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n.

Prove that

n\le 16

and determine the number of

n

-tuples

(a_1,a_2,\ldots,a_n

) satisfying these conditions for

n=16

5