A304032 - OEIS

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 #22 May 07 2018 03:50:53

%S 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 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 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 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 The even number 58958 cannot be written as p + 2^k + 3^m with p and 2^k + 3^m both prime.

%C Clearly, a(n) <= A303702(n). We note that a(n) > 0 for all n = 2..5*10^8.

%C See also A304034 for a related conjecture.

%D 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 Zhi-Wei Sun, <a href="/A304032/b304032.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://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 a(3) = 1 since 2*3 = 3 + 2^1 + 3^0 with 3 = 2^1 + 3^0 prime.

%t qq[n_]:=qq[n]=SquareFreeQ[n]&&Length[FactorInteger[n]]<=2;

%t 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 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 nonn

%O 1,4

%A _Zhi-Wei Sun_, May 04 2018