A303998 Number of ways to write 2*n+1 as p + 2^k + binomial(2*m,m), where p is a prime, and k and m are positive integers.

0, 0, 1, 2, 3, 4, 4, 5, 3, 6, 5, 6, 8, 7, 5, 7, 7, 6, 8, 11, 5, 8, 9, 5, 10, 8, 7, 8, 7, 5, 7, 10, 6, 9, 9, 5, 11, 12, 8, 13, 12, 9, 8, 15, 9, 11, 12, 11, 7, 10, 9, 10, 14, 9, 12, 12, 11, 11, 12, 9, 9, 12, 8, 5, 13, 9, 10, 14, 10, 13, 9, 15, 10, 12, 9, 12, 11, 9, 11, 13

Offset

1,4

Comments

Conjecture: a(n) > 0 for all n > 2.

This has been verified for n up to 10^9.

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(3) = 1 since 2*3+1 = 3 + 2^1 + binomial(2*1,1) with 3 prime.
a(4) = 2 since 2*4+1 = 3 + 2^2 + binomial(2*1,1) = 5 + 2^1 + binomial(2*1,1) with 3 and 5 both prime.

Mathematica

c[n_]:=c[n]=Binomial[2n, n];
tab={}; Do[r=0; k=1; Label[bb]; If[c[k]>2n, Goto[aa]]; Do[If[PrimeQ[2n+1-c[k]-2^m], r=r+1], {m, 1, Log[2, 2n+1-c[k]]}]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

Crossrefs

Cf. A000040, A000079, 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, A303997, A304031.

Sequence in context: A334049 A058277 A065852 * A319712 A319715 A088807

Adjacent sequences: A303995 A303996 A303997 * A303999 A304000 A304001

Keyword

nonn

Author

Zhi-Wei Sun, May 04 2018

Status

approved