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

1,2

Comments

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

Links

Table of n, a(n) for n=1..6.

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.

Example

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.

Crossrefs

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

Sequence in context: A287448 A325235 A135116 * A259559 A050187 A101506

Adjacent sequences: A362740 A362741 A362742 * A362744 A362745 A362746

Keyword

nonn,more

Author

Zhi-Wei Sun, May 01 2023

Status

approved