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A304034 Number of ways to write n as p + 2^k + (1+(n mod 2))*3^m with p prime, where k and m are positive integers with 2^k + (1+(n mod 2))*3^m squarefree.

%I A304034 #22 May 19 2018 09:42:16
%S A304034 0,0,0,0,0,0,0,1,0,2,0,2,1,3,1,4,2,5,1,3,2,5,1,7,3,3,4,4,4,6,2,3,5,6,
%T A304034 2,7,3,5,5,6,5,9,3,4,6,7,2,12,2,5,6,7,4,10,3,3,5,8,2,8,3,4,6,8,5,9,4,
%U A304034 2,7,7,3,13,5,5,9,7,5,13,3,6,10,7,5,10,5,7,7,9,8,13
%N A304034 Number of ways to write n as p + 2^k + (1+(n mod 2))*3^m with p prime, where k and m are positive integers with 2^k + (1+(n mod 2))*3^m squarefree.
%C A304034 Conjecture: a(n) > 0 for all n > 11.
%C A304034 This has been verified for n up to 10^10.
%C A304034 See also A304081 for a similar conjecture.
%H A304034 Zhi-Wei Sun, <a href="/A304034/b304034.txt">Table of n, a(n) for n = 1..10000</a>
%H A304034 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 A304034 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 A304034 a(8) = 1 since 8 = 3 + 2^1 + 3^1 with 3 prime and 2^1 + 3^1 = 5 squarefree.
%e A304034 a(13) = 1 since 13 = 3 + 2^2 + 2*3^1 with 3 prime and 2^2 + 2*3^1 = 2*5 squarefree.
%e A304034 a(19) = 1 since 19 = 5 + 2^3 + 2*3^1 with 5 prime and 2^3 + 2*3^1 = 2*7 squarefree.
%e A304034 a(23) = 1 since 23 = 13 + 2^2 + 2*3^1 with 13 prime and 2^2 + 2*3 = 2*5 squarefree.
%t A304034 tab={};Do[r=0;Do[If[SquareFreeQ[2^k+(1+Mod[n,2])*3^m]&&PrimeQ[n-2^k-(1+Mod[n,2])*3^m],r=r+1],{k,1,Log[2,n]},{m,1,If[2^k==n,-1,Log[3,(n-2^k)/(1+Mod[n,2])]]}];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y A304034 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, A304032, A304081.
%K A304034 nonn
%O A304034 1,10
%A A304034 _Zhi-Wei Sun_, May 06 2018