This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A303821 #24 May 10 2018 08:25:37
%S A303821 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 A303821 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 A303821 6,5,12,7,10,15,8,15,10,11,13,11,7,9,8,9,12,14
%N A303821 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 A303821 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 A303821 This has been verified for n up to 10^10.
%C A303821 See also A303934 and A304081 for further refinements, and A303932 and A304034 for similar conjectures.
%H A303821 Zhi-Wei Sun, <a href="/A303821/b303821.txt">Table of n, a(n) for n = 1..10000</a>
%H A303821 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 A303821 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 A303821 a(2) = 1 since 2*2 = 2 + 2^0 + 5^0 with 2 prime.
%e A303821 a(3) = 1 since 2*3 = 3 + 2^1 + 5^0 with 3 prime.
%e A303821 a(5616) = 2 since 2*5616 = 9059 + 2^11 + 5^3 = 10979 + 2^7 + 5^3 with 9059 and 10979 both prime.
%t A303821 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 A303821 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 A303821 nonn
%O A303821 1,4
%A A303821 _Zhi-Wei Sun_, May 01 2018