A303997 - OEIS

%I #14 May 04 2018 09:04:36

%S 0,1,2,2,3,4,4,4,4,4,5,4,6,6,4,6,8,6,5,8,5,5,8,5,6,10,4,4,7,5,5,7,6,4,

%T 8,4,6,11,6,5,10,8,7,9,11,7,10,7,4,11,9,9,9,10,8,12,9,9,11,9,5,8,8,4,

%U 11,8,7,8,8,7,10,8,7,6,7,5,10,9,7,12,8,5,7,9,8,9,8,6,8,11

%N Number of ways to write 2*n as p + 3^k + binomial(2*m,m), where p is a prime, and k and m are nonnegative integers.

%C   502743678 is the first value of n > 1 with a(n) = 0.

%H Zhi-Wei Sun, <a href="/A303997/b303997.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 + 3^0 + binomial(2*0,0) with 2 prime.

%e a(3) = 2 since 2*3 = 3 + 3^0 + binomial(2*1,1) = 2 + 3^1 + binomial(2*0,0) with 3 and 2 both prime.

%t c[n_]:=c[n]=Binomial[2n,n];

%t tab={};Do[r=0;k=0;Label[bb];If[c[k]>=2n,Goto[aa]];Do[If[PrimeQ[2n-c[k]-3^m],r=r+1],{m,0,Log[3,2n-c[k]]}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,90}];Print[tab]

%Y Cf. A000040, A000224, A000984, 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, A303998.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, May 04 2018