A226809 -id:A226809 - cp's OEIS frontend

A226790 Numbers of the form (3^j + 5^k)/2, for j and k >= 0.

1, 2, 3, 4, 5, 7, 13, 14, 16, 17, 26, 41, 43, 53, 63, 64, 67, 76, 103, 122, 124, 134, 184, 313, 314, 317, 326, 353, 365, 367, 377, 427, 434, 677, 1094, 1096, 1106, 1156, 1406, 1563, 1564, 1567, 1576, 1603, 1684, 1927, 2656, 3281, 3283, 3293, 3343, 3593

Offset: 1

T. D. Noe, Jun 19 2013

nonn

Cf. A226809 (3^j + 5^k).

a = 3; b = 5; mx = 8000; Union[Flatten[Table[a^n + b^m, {m, 0, Log[b, mx]}, {n, 0, Log[a, mx - b^m]}]]]/2

A362861 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,...}.

Original entry on oeis.org

2, 7, 10, 11, 12, 27, 35, 50, 51, 52, 55, 60, 135, 255

Offset: 1

Zhi-Wei Sun, May 05 2023

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.
a(15) >= 10^6. - Martin Ehrenstein, May 16 2023

			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).

Cf. A055235, A055237, A226809, A226816, A362743.

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A226790 Numbers of the form (3^j + 5^k)/2, for j and k >= 0.

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A362861 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,...}.

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