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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| On Dec 23, 2008, Zhi-Wei Sun made a conjecture which 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. 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 (zwsun(AT)nju.edu.cn) has offered a monetary reward for settling this conjecture.
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REFERENCES
| 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.
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
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LINKS
| Zhi-Wei SUN, Table of n, a(n), n=1..50000.
D. S. 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 (II)
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EXAMPLE
| 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
| 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
| Cf. A000040, A000045. See A144559 for another version.
Sequence in context: A206710 A175502 A075527 * A060019 A093451 A106006
Adjacent sequences: A154254 A154255 A154256 * A154258 A154259 A154260
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KEYWORD
| nice,nonn
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AUTHOR
| Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 05 2009
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EXTENSIONS
| The new verification record is 10^14 (due to D. McNeil).---Zhi-Wei Sun Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 17 2009
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