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A303934 Number of ways to write 2*n as p + 2^k + 5^m with p prime and 2^k + 5^m squarefree, where k and m are nonnegative integers.

%I A303934 #15 May 07 2018 03:36:12
%S A303934 0,1,1,3,3,2,2,3,3,4,3,5,4,4,3,4,5,7,4,7,4,8,7,6,7,6,5,5,5,7,5,8,5,5,
%T A303934 8,6,9,9,6,8,6,6,7,8,4,7,8,7,3,10,6,7,8,7,7,9,5,8,7,6,5,5,6,3,11,7,9,
%U A303934 12,8,12,10,11,11,9,7,9,7,8,8,11,7,11,8,9,15,11,8,9,8,9
%N A303934 Number of ways to write 2*n as p + 2^k + 5^m with p prime and 2^k + 5^m squarefree, where k and m are nonnegative integers.
%C A303934 Conjecture: a(n) > 0 for all n > 1.
%C A303934 This has been verified for all n = 2..10^10.
%C A303934 Note that a(n) <= A303821(n).
%H A303934 Zhi-Wei Sun, <a href="/A303934/b303934.txt">Table of n, a(n) for n = 1..10000</a>
%H A303934 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 A303934 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 A303934 a(2) = 1 since 2*2 = 2 + 2^0 + 5^0 with 2 prime and 2^0 + 5^0 squarefree.
%e A303934 a(3) = 1 since 2*3 = 3 + 2^1 + 5^0 with 3 prime and 2^1 + 5^0 squarefree.
%t A303934 tab={};Do[r=0;Do[If[SquareFreeQ[2^k+5^m]&&PrimeQ[2n-2^k-5^m],r=r+1],{k,0,Log[2,2n-1]},{m,0,Log[5,2n-2^k]}];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y A303934 Cf. A000040, A000079, A000351, A005117, A118955, 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, A304034, A304081, A304122.
%K A303934 nonn
%O A303934 1,4
%A A303934 _Zhi-Wei Sun_, May 03 2018