1,8

On Jan 09, 2009, Zhi-Wei Sun conjectured that a(n)>0 for all n=6,7,.... ; in other words, any integer n>5 can be written in the form p+F_s+L_{3t} with p an odd prime, s positive and t nonnegative. [Compare this with the conjecture related to the sequence A154290.] Sun verified the above conjecture up to 5*10^6 and Qing-Hu Hou continued the verification up to 2*10^8. If we set v_0=2, v_1=4 and v_{n+1}=4v_n+v_{n-1} for n=1,2,3,..., then L_{3t}=v_t is at least 4^t for every t=0,1,2,.... On Jan 17, 2009 Douglas McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.

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 summary concerning my conjecture n=p+F_s+F_t

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

a(n)=|{<p,s,t>: p+F_s+L_{3t}=n with p an odd prime, s>1 and t nonnegative}|

For n=8 the a(8)=3 solutions are 3+F_4+L_0, 3+F_2+L_3, 5+F_2+L_0.

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

Cf. A000040, A000045, A000032, A154257, A154285, A154290, A154364, A154417

Sequence in context: A074830 A127750 A112528 this_sequence A057034 A075410 A023513

Adjacent sequences: A154418 A154419 A154420 this_sequence A154422 A154423 A154424

nice,nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 09 2009

Added McNeil's counterexample. - Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 20 2009