A226808 -id:A226808 - cp's OEIS frontend

A362743 Positive integers which cannot be written as a sum of distinct numbers of the form 4^a + 5^b (a,b >= 0).

1, 3, 4, 10, 12, 18

Offset: 1

Zhi-Wei Sun, May 01 2023

If a(7) exists, it will be greater than 2750.
Conjecture 1: The only terms of the current sequence are 1, 3, 4, 10, 12, 18. Moreover, any positive integer not among 1, 3, 4, 8, 10, 12, 13, 18, 25, 39, 42 can be written as a sum of numbers of the form 4^a + 5^b (a,b>=0) with no one summand dividing another.
Conjecture 2: Let k and m be positive integers greater than one with k*m even. Then, any sufficiently large integer n can be written as a sum of distinct numbers of the form k^a + m^b with a and b nonnegative integers.
Conjecture 3: Let k and m be positive integers greater than one with k*m even. Then, any sufficiently large integer n can be written as a sum of 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 A362861 for similar conjectures.
a(7) > 50000. - Martin Ehrenstein, May 16 2023

			a(1) = 1 since 4^a + 5^b > 1 for all a,b >= 0.
a(2) = 3 since 2 = 4^0 + 5^0, and 3 cannot be written as a sum of distinct numbers of the form 4^a + 5^b with a,b >= 0.

B. J. Birch, Note on a problem of Erdos, Proc. Cambridge Philos. Soc. 55 (1959), 370-373.
P. Erdos and M. Lewin, d-complete sequences of integers, Math. Comp. 65 (1996), 837--840.

Cf. A226806, A226807, A226808, A226810, A226812, A362861.

cp's OEIS Frontend

A362743 Positive integers which cannot be written as a sum of distinct numbers of the form 4^a + 5^b (a,b >= 0).

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