

A154421


Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.


2



0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 2, 5, 2, 4, 3, 4, 4, 5, 2, 6, 2, 7, 5, 7, 3, 9, 3, 9, 4, 7, 3, 6, 4, 9, 3, 10, 3, 8, 4, 6, 5, 8, 6, 8, 3, 9, 4, 8, 6, 8
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OFFSET

1,8


COMMENTS

On Jan 09 2009, ZhiWei 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 QingHu 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_{n1} 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, D. S. McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.


REFERENCES

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103107.


LINKS

K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m2^n and x^2F_{3n}/2, Math. Comp. 78 (2009) 1853, [DOI], arXiv:math.NT/0702382


FORMULA

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


EXAMPLE

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


MATHEMATICA

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n2*Fibonacci[3x+1]+Fibonacci[3x]Fibonacci[y]], 1, 0], {x, 0, Log[2, n]}, {y, 2, 2*Log[2, Max[2, n2*Fibonacci[3x+1]+Fibonacci[3x]]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

McNeil's counterexample added by ZhiWei Sun, Jan 20 2009


STATUS

approved



