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A154421
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Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.
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2
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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
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1,8
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COMMENTS
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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, D. S. McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.
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REFERENCES
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R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
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LINKS
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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. 78 (2009) 1853, [DOI], arXiv:math.NT/0702382
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FORMULA
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a(n) = |{<p,s,t>: p+F_s+L_{3t}=n with p an odd prime, s>1 and t nonnegative}|.
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EXAMPLE
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For n=8 the a(8)=3 solutions are 3 + F_4 + L_0, 3 + F_2 + L_3, 5 + F_2 + L_0.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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McNeil's counterexample added by Zhi-Wei Sun, Jan 20 2009
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STATUS
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approved
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