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,...}.

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

Offset

1,1

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.

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

Links

Table of n, a(n) for n=1..14.

Example

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).

Crossrefs

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

Sequence in context: A319932 A236243 A024831 * A194421 A043357 A023730

Adjacent sequences: A362858 A362859 A362860 * A362862 A362863 A362864

Keyword

nonn,more

Author

Zhi-Wei Sun, May 05 2023

Status

approved