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A304032 Number of ways to write 2*n as p + 2^k + 3^m with p prime and 2^k + 3^m a product of at most two distinct primes, where k and m are nonnegative integers.

%I A304032 #22 May 07 2018 03:50:53
%S A304032 0,1,1,3,4,4,4,6,6,5,8,9,4,6,7,4,9,10,6,9,10,6,11,14,7,9,11,5,10,9,6,
%T A304032 12,10,3,11,15,7,12,16,7,9,14,9,12,14,8,12,16,5,12,18,10,12,16,9,12,
%U A304032 19,10,13,17,6,10,15,6,10,16,10,12,15,10,17,20,8,14,15,8,11,18,9,12
%N A304032 Number of ways to write 2*n as p + 2^k + 3^m with p prime and 2^k + 3^m a product of at most two distinct primes, where k and m are nonnegative integers.
%C A304032 The even number 58958 cannot be written as p + 2^k + 3^m with p and 2^k + 3^m both prime.
%C A304032 Clearly, a(n) <= A303702(n). We note that a(n) > 0 for all n = 2..5*10^8.
%C A304032 See also A304034 for a related conjecture.
%D A304032 J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16(1973), 157-176.
%H A304032 Zhi-Wei Sun, <a href="/A304032/b304032.txt">Table of n, a(n) for n = 1..10000</a>
%H A304032 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 A304032 Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_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 A304032 a(3) = 1 since 2*3 = 3 + 2^1 + 3^0 with 3 = 2^1 + 3^0 prime.
%t A304032 qq[n_]:=qq[n]=SquareFreeQ[n]&&Length[FactorInteger[n]]<=2;
%t A304032 tab={};Do[r=0;Do[If[qq[2^k+3^m]&&PrimeQ[2n-2^k-3^m],r=r+1],{k,0,Log[2,2n-1]},{m,0,Log[3,2n-2^k]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y A304032 Cf. A000040, A000079, A000224, A005117, A118955, A155216, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702,  A303821, A303932, A303934, A303949, A304031, A304034, A304081.
%K A304032 nonn
%O A304032 1,4
%A A304032 _Zhi-Wei Sun_, May 04 2018