
COMMENTS

If a(15) exists, it should be greater than 10290.
Conjecture 1: (i) The current sequence only has the listed 14 terms. Also, each positive even number can be written as a sum of distinct elements of the set {3^a + 3^b: a,b = 0,1,2,...}.
(ii) Each positive even number can be written as a sum of distinct elements of the set {3^a + 7^b: a,b = 0,1,2,...}. Also, any positive even number not equal to 12 can be written as a sum of numbers of the form 3^a + 5^b (a,b >= 0) with no summand dividing another.
Conjecture 2: Let k and m be positive odd numbers greater than one. Then, any sufficiently large even numbers can be written as a sum of distinct elements of the set {k^a + m^b: a,b = 0,1,2,...}.
Conjecture 3: Let k and m be positive odd numbers greater than one. Then, any sufficiently large even numbers can be written as a sum of some numbers of the form k^a + m^b (a,b >= 0) with no summand dividing another.
Clearly, Conjecture 3 is stronger than Conjecture 2.
See also A362743 for similar conjectures.
