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%I #24 May 10 2018 08:25:37
%S 0,1,1,3,3,4,4,5,3,6,5,5,6,6,4,7,6,7,7,10,4,9,10,6,10,8,5,8,6,7,7,9,5,
%T 8,11,6,10,11,6,11,8,6,8,11,4,9,9,7,6,11,6,7,11,7,10,11,5,11,9,6,7,6,
%U 6,5,12,7,10,15,8,15,10,11,13,11,7,9,8,9,12,14
%N Number of ways to write 2*n as p + 2^x + 5^y, where p is a prime, and x and y are nonnegative integers.
%C Conjecture: a(n) > 0 for all n > 1. Moreover, for any integer n > 4, we can write 2*n as p + 2^x + 5^y, where p is an odd prime, and x and y are positive integers.
%C This has been verified for n up to 10^10.
%C See also A303934 and A304081 for further refinements, and A303932 and A304034 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A303821/b303821.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(2) = 1 since 2*2 = 2 + 2^0 + 5^0 with 2 prime.
%e a(3) = 1 since 2*3 = 3 + 2^1 + 5^0 with 3 prime.
%e a(5616) = 2 since 2*5616 = 9059 + 2^11 + 5^3 = 10979 + 2^7 + 5^3 with 9059 and 10979 both prime.
%t tab={};Do[r=0;Do[If[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,1,80}];Print[tab]
%Y Cf. A000040, A000079, A000351, 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, A303932, A303934, A304034, A304081.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, May 01 2018
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