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%I #12 May 16 2023 16:12:38
%S 2,7,10,11,12,27,35,50,51,52,55,60,135,255
%N Positive integers n such that 2*n cannot be written as a sum of distinct elements of the set {5^a + 5^b: a,b = 0,1,2,...}.
%C If a(15) exists, it should be greater than 10290.
%C 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,...}.
%C (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.
%C 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,...}.
%C 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.
%C Clearly, Conjecture 3 is stronger than Conjecture 2.
%C See also A362743 for similar conjectures.
%C a(15) >= 10^6. - _Martin Ehrenstein_, May 16 2023
%e a(1) = 2, since 2*1 = 5^0 + 5^0 but 2*2 cannot be written as a sum of distinct numbers of the form 5^a + 5^b (a,b >= 0).
%Y Cf. A055235, A055237, A226809, A226816, A362743.
%K nonn,more
%O 1,1
%A _Zhi-Wei Sun_, May 05 2023
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