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%I #28 Aug 25 2023 17:21:03
%S 0,1,2,3,4,5,5,7,6,6,9,9,5,8,9,6,9,11,8,10,11,7,12,15,8,10,12,7,10,9,
%T 8,12,11,5,12,16,7,13,17,8,10,15,10,13,14,10,12,17,7,12,18,11,13,17,
%U 10,13,20,11,14,17,8,10,16,7,10
%N Number of ways to write 2*n as p + 2^k + 3^m, where p is a prime, and k and m are nonnegative integers.
%C Conjecture: a(n) > 0 for all n > 1. In other words, any even number greater than 2 can be written as the sum of a prime, a power of 2 and a power of 3.
%C It has been verified that a(n) > 0 for all n = 2..3*10^9.
%C a(n) > 0 for n <= 10^11. - _Jud McCranie_, Jun 25 2023
%C a(n) > 0 for n < 10^12. - _Jud McCranie_, Jul 11 2023
%C a(n) > 0 for n <= 4*10^12. - _Jud McCranie_, Aug 17 2023
%H Zhi-Wei Sun, <a href="/A303702/b303702.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/116f.pdf">Mixed sums of primes and other terms</a>, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
%H Zhi-Wei Sun, <a href="https://link.springer.com/chapter/10.1007%2F978-3-319-68032-3_20">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e a(2) = 1 since 2*2 = 2 + 2^0 + 3^0 with 2 prime.
%e a(3) = 2 since 2*3 = 2 + 2^0 + 3^1 = 3 + 2^1 + 3^0 with 2 and 3 prime.
%t tab={};Do[r=0;Do[If[PrimeQ[2n-2^x-3^y],r=r+1],{x,0,Log[2,2n-1]},{y,0,Log[3,2n-2^x]}];tab=Append[tab,r],{n,1,65}];Print[tab]
%Y Cf. A000040, A000079, A000244, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 29 2018
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