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A305232 Number of ordered ways to write 2*n+1 as p + binomial(2k,k) + 2*binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.

%I A305232 #23 Dec 04 2024 16:09:51
%S A305232 0,0,1,2,3,3,3,4,3,5,4,5,6,5,4,4,6,6,4,5,4,6,6,6,7,6,6,4,5,6,6,8,5,5,
%T A305232 6,5,7,9,8,5,8,9,6,9,7,8,6,6,4,7,8,7,7,4,8,10,9,7,8,9,5,7,6,5,7,7,7,3,
%U A305232 6,7,7,9,6,9,6,9,9,7,7,8,9,6,5,8,10,10,6,8,7,9
%N A305232 Number of ordered ways to write 2*n+1 as p + binomial(2k,k) + 2*binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.
%C A305232 The first value of n > 2 with a(n) = 0 is 15212443837. Neither 2*15212443837 + 1 = 30424887675 nor 2*15657981007 + 1 = 31315962015 can be written as the sum of a prime, a central binomial coefficient and twice a central binomial coefficient.
%H A305232 Zhi-Wei Sun, <a href="/A305232/b305232.txt">Table of n, a(n) for n = 1..100000</a>
%H A305232 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/116f.pdf">Mixed sums of primes and other terms</a>, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
%H A305232 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 A305232 a(3) = 1 since 2*3 + 1 = 7 = 3 + binomial(2*1,1) + 2*binomial(2*0,0) with 3 an odd prime.
%e A305232 a(368233372) = 1 since 2*368233372 + 1 = 736466745 = 735761311 + binomial(2*11,11) + 2*binomial(2*0,0) with 735761311 an odd prime.
%e A305232 a(5274658504) = 1 since 2*5274658504 + 1 = 10549317009 = 10549316083 + binomial(2*6,6) + 2*binomial(2*0,0) with 10549316083 an odd prime.
%e A305232 a(8722422187) = 1 since 2*8722422187 + 1 = 17444844375 = 17444844367 + binomial(2*2,2) + 2*binomial(2*0,0) with 17444844367 an odd prime.
%e A305232 a(10296844792) = 1 since 2*10296844792 + 1 = 20593689585 = 20593688659 + binomial(2*6,6) + 2*binomial(2*0,0) with 20593688659 an odd prime.
%t A305232 tab={};Do[r=0;k=0;Label[aa];k=k+1;If[Binomial[2k,k]>=2n+1`,Goto[cc]];m=0;Label[bb];If[2*Binomial[2m,m]>=2n+1-Binomial[2k,k],Goto[aa]]; If[PrimeQ[2n+1-Binomial[2k,k]-2*Binomial[2m,m]],r=r+1];m=m+1;Goto[bb];Label[cc];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y A305232 Cf. A000040, A000984, A303540, A303656, A303702, A303821, A303934, A304034, A304081, A305030.
%K A305232 nonn
%O A305232 1,4
%A A305232 _Zhi-Wei Sun_, May 27 2018