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

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Last modified June 12 17:05 EDT 2024. Contains 373334 sequences. (Running on oeis4.)