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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, 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
Sequence in context: A368700 A095395 A029134 * A276646 A029130 A081611
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 04 2018
STATUS
approved

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Last modified June 12 17:08 EDT 2024. Contains 373339 sequences. (Running on oeis4.)