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Positive integers which cannot be written as a sum of distinct numbers of the form 4^a + 5^b (a,b >= 0).
1

%I #35 Nov 02 2023 14:19:16

%S 1,3,4,10,12,18

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

%C If a(7) exists, it will be greater than 2750.

%C 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.

%C 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.

%C 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.

%C Clearly, Conjecture 3 is stronger than Conjecture 2.

%C See also A362861 for similar conjectures.

%C a(7) > 50000. - _Martin Ehrenstein_, May 16 2023

%H B. J. Birch, <a href="https://doi.org/10.1017/S0305004100034150">Note on a problem of Erdos</a>, Proc. Cambridge Philos. Soc. 55 (1959), 370-373.

%H P. Erdos and M. Lewin, <a href="https://doi.org/10.1090/S0025-5718-96-00707-7">d-complete sequences of integers</a>, Math. Comp. 65 (1996), 837--840.

%e a(1) = 1 since 4^a + 5^b > 1 for all a,b >= 0.

%e 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.

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

%K nonn,more

%O 1,2

%A _Zhi-Wei Sun_, May 01 2023