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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.
5

%I #33 May 07 2018 03:51:09

%S 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 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 5,10,9,8,5,9,5,7,10,5,4,10,7,6,8,6,7,8,7,6,8,6

%N 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 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 See also A303934 and A304081 for related conjectures.

%H Zhi-Wei Sun, <a href="/A303949/b303949.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(3) = 1 since 2*3+1 = 3 + 2*(2^0+5^0) with 3 prime.

%t qq[n_]:=qq[n]=SquareFreeQ[n]&&Length[FactorInteger[n]]<=3;

%t 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 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 nonn

%O 1,4

%A _Zhi-Wei Sun_, May 05 2018