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.
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, 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, 6, 7, 7, 9, 6, 9, 6, 9, 9, 7, 7, 8, 9, 6, 5, 8, 10, 10, 6, 8, 7, 9
Offset: 1
Keywords
Examples
a(3) = 1 since 2*3 + 1 = 7 = 3 + binomial(2*1,1) + 2*binomial(2*0,0) with 3 an odd prime. a(368233372) = 1 since 2*368233372 + 1 = 736466745 = 735761311 + binomial(2*11,11) + 2*binomial(2*0,0) with 735761311 an odd prime. a(5274658504) = 1 since 2*5274658504 + 1 = 10549317009 = 10549316083 + binomial(2*6,6) + 2*binomial(2*0,0) with 10549316083 an odd prime. a(8722422187) = 1 since 2*8722422187 + 1 = 17444844375 = 17444844367 + binomial(2*2,2) + 2*binomial(2*0,0) with 17444844367 an odd prime. a(10296844792) = 1 since 2*10296844792 + 1 = 20593689585 = 20593688659 + binomial(2*6,6) + 2*binomial(2*0,0) with 20593688659 an odd prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
- Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
- Zhi-Wei Sun, Conjectures on representations involving primes, 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 arXiv:1211.1588 [math.NT], 2012-2017.)
Crossrefs
Programs
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Mathematica
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]
Comments