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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. 2
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.

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(2) = 1 since 2*2 = 2 + 3^0 + binomial(2*0,0) with 2 prime.

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.

MATHEMATICA

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

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]

CROSSREFS

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.

Sequence in context: A175214 A095395 A029134 * A276646 A029130 A081611

Adjacent sequences:  A303994 A303995 A303996 * A303998 A303999 A304000

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 04 2018

STATUS

approved

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Last modified April 3 19:43 EDT 2020. Contains 333198 sequences. (Running on oeis4.)