cp's OEIS Frontend

A303997 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.

%I A303997 #14 May 04 2018 09:04:36
%S A303997 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 A303997 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 A303997 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 A303997 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 A303997   502743678 is the first value of n > 1 with a(n) = 0.
%H A303997 Zhi-Wei Sun, <a href="/A303997/b303997.txt">Table of n, a(n) for n = 1..10000</a>
%H A303997 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 A303997 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 A303997 a(2) = 1 since 2*2 = 2 + 3^0 + binomial(2*0,0) with 2 prime.
%e A303997 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 A303997 c[n_]:=c[n]=Binomial[2n,n];
%t A303997 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 A303997 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 A303997 nonn
%O A303997 1,3
%A A303997 _Zhi-Wei Sun_, May 04 2018