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A303949 Number of ways to write 2*n+1 as p + 2*(2^k+5^m) with p prime and 2^k+5^m a product of at most three distinct primes, where k and m are nonnegative integers.

%I A303949 #33 May 07 2018 03:51:09
%S A303949 0,0,1,2,2,2,3,5,3,3,4,3,3,4,3,4,3,4,3,4,4,5,5,4,4,5,5,6,4,3,6,7,3,6,
%T A303949 9,7,5,8,7,6,7,9,7,8,2,8,9,5,5,6,6,7,6,6,7,10,6,7,9,5,6,8,6,3,6,7,7,8,
%U A303949 5,10,9,8,5,9,5,7,10,5,4,10,7,6,8,6,7,8,7,6,8,6
%N A303949 Number of ways to write 2*n+1 as p + 2*(2^k+5^m) with p prime and 2^k+5^m a product of at most three distinct primes, where k and m are nonnegative integers.
%C A303949 4787449 is the first value of n > 2 with a(n) = 0, and 2*4787449+1 = 9574899 has the unique representation as p + 2*(2^k+5^m): 9050609 + 2*(2^18+5^0) with 9050609 prime and 2^18+5^0 = 5*13*37*109.
%C A303949 See also A303934 and A304081 for related conjectures.
%H A303949 Zhi-Wei Sun, <a href="/A303949/b303949.txt">Table of n, a(n) for n = 1..10000</a>
%H A303949 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 A303949 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 A303949 a(3) = 1 since 2*3+1 = 3 + 2*(2^0+5^0) with 3 prime.
%t A303949 qq[n_]:=qq[n]=SquareFreeQ[n]&&Length[FactorInteger[n]]<=3;
%t A303949 tab={};Do[r=0;Do[If[SquareFreeQ[2^k+5^m]&&PrimeQ[2n+1-2(2^k+5^m)],r=r+1],{k,0,Log[2,n]},{m,0,Log[5,n+1/2-2^k]}];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y A303949 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, A303934, A304031, A304032, A304081.
%K A303949 nonn
%O A303949 1,4
%A A303949 _Zhi-Wei Sun_, May 05 2018