1,6

Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a Lucas number, with the Fibonacci number or the Lucas number odd. Moreover, Sun conjectured that lim inf_n a(n)/log(n) is greater than 3 and smaller than 4.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

T. Tao, A remark on primality testing and decimal expansions, J. Austral. Math. Soc., in press. arXiv:0802.3361

K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp., in press. arXiv:math.NT/0702382

Zhi-Wei SUN, Table of n, a(n), n=1..50000.

Douglas McNeil, Sun's strong conjecture

Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t

Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t

For n=10 the a(10)=7 solutions are 3+F_4+L_3, 3+F_5+L_0, 5+F_2+L_3, 5+F_3+L_2, 5+F_4+L_0, 7+F_2+L_0, 7+F_3+L_1.

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n, 2]==0||Mod[x, 3]>0)&&PQ[n-(2*Fibonacci[x+1]-Fibonacci[x])-Fibonacci[y]], 1, 0], {x, 0, 2*Log[2, n]}, {y, 2, 2*Log[2, Max[2, n-(2*Fibonacci[x+1]-Fibonacci[x])]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]

Cf. A000040, A000045, A000032, A154257, A154285

Sequence in context: A066113 A163831 A081836 this_sequence A002334 A115732 A048947

Adjacent sequences: A154287 A154288 A154289 this_sequence A154291 A154292 A154293

nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 06 2009, Jan 07 2008