This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A362743 #35 Nov 02 2023 14:19:16 %S A362743 1,3,4,10,12,18 %N A362743 Positive integers which cannot be written as a sum of distinct numbers of the form 4^a + 5^b (a,b >= 0). %C A362743 If a(7) exists, it will be greater than 2750. %C A362743 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 A362743 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 A362743 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 A362743 Clearly, Conjecture 3 is stronger than Conjecture 2. %C A362743 See also A362861 for similar conjectures. %C A362743 a(7) > 50000. - _Martin Ehrenstein_, May 16 2023 %H A362743 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 A362743 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 A362743 a(1) = 1 since 4^a + 5^b > 1 for all a,b >= 0. %e A362743 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 A362743 Cf. A226806, A226807, A226808, A226810, A226812, A362861. %K A362743 nonn,more %O A362743 1,2 %A A362743 _Zhi-Wei Sun_, May 01 2023