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A154257
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Number of triples <p,s,t> such that p+F_s+F_t=n, where p is an odd prime, s and t are greater than one and the Fibonacci number F_s or F_t is odd.
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20
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0, 0, 0, 0, 1, 2, 3, 4, 6, 6, 7, 6, 7, 8, 6, 10, 8, 10, 10, 10, 12, 10, 10, 10, 12, 14, 13, 12, 15, 8, 12, 12, 13, 14, 13, 10, 16, 10, 13, 16, 11, 16, 11, 14, 17, 16, 15, 12, 12, 16, 11, 20, 13, 14, 13, 12, 12, 18, 12, 16, 14, 14, 19, 16, 18, 20, 16, 18, 15, 18, 16, 12, 16, 18, 19, 22, 18
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OFFSET
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1,6
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COMMENTS
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On Dec 23 2008, _Zhi-Wei_ Sun made a conjecture that states that a(n)>0 for all n=5,6,... (i.e., any integer n>4 can be written as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number). This has been verified for n up to 10^14 by D. S. McNeil; the conjecture looks more difficult than the Goldbach conjecture since Fibonacci numbers are much more sparse than prime numbers. Sun also conjectured that c=lim inf_n a(n)/log n is greater than 2 and smaller than 3.
Zhi-Wei Sun has offered a monetary reward for settling this conjecture.
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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.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.
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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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EXAMPLE
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For n=9 the a(9)=6 solutions are 3 + F_4 + F_4, 3 + F_2 + F_5, 3 + F_5 + F_2, 5 + F_3 + F_3, 5 + F_2 + F_4, 5 + F_4 + F_2.
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MATHEMATICA
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PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n, 2]==0||Mod[x, 3]>0)&&PQ[n-Fibonacci[x]-Fibonacci[y]], 1, 0], {x, 2, 2*Log[2, Max[2, n]]}, {y, 2, 2*Log[2, Max[2, n-Fibonacci[x]]]}] 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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STATUS
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approved
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