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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
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.)
EXAMPLE
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.
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]
CROSSREFS
Sequence in context: A175239 A176228 A322822 * A322974 A326201 A342625
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 27 2018
STATUS
approved

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Last modified July 21 04:02 EDT 2024. Contains 374463 sequences. (Running on oeis4.)