OFFSET
1,6
COMMENTS
On Jan 16 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=5,6,... and verified this up to 5*10^6. (Sun also thought that lim inf_n a(n)/log(n) is a positive constant.) D. S. McNeil continued the verification up to 10^13 and found no counterexamples. The conjecture is similar to a conjecture of Qing-Hu Hou and Jiang Zeng related to the sequence A154404; both conjectures were motivated by Sun's recent conjecture on sums of primes and Fibonacci numbers (cf. A154257).
REFERENCES
R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge Univ. Press, 1999, Chapter 6.
LINKS
Zhi-Wei Sun, Table of n, a(n), n=1..100000
D. S. McNeil, Sun's strong conjecture
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Z.-W. Sun and R. Tauraso, Congruences involving Catalan numbers, arXiv:0709.1665.
FORMULA
a(n) = |{<p,s,t>: p+L_s+C_t=n with p an odd prime, s>=0 and t>0}|.
EXAMPLE
For n=10 the a(10)=5 solutions are 3 + L_0 + C_3, 5 + L_2 + C_2, 5 + L_3 + C_1, 7 + L_0 + C_1, 7 + L_1 + C_2.
MATHEMATICA
PQ[m_]:=m>2&&PrimeQ[m] L[x_]:=2*Fibonacci[x+1]-Fibonacci[x] RN[n_]:=Sum[If[PQ[n-L[x]-CatalanNumber[y]], 1, 0], {x, 0, 2*Log[2, n]}, {y, 1, 2*Log[2, Max[2, n-L[x]+1]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 100000}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 17 2009
EXTENSIONS
More terms (from b-file) added by N. J. A. Sloane, Aug 31 2009
STATUS
approved