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Search: id:A155114
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| A155114 |
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Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number |
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+0
8
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| 0, 0, 0, 0, 0, 1, 1, 3, 2, 6, 3, 7, 3, 8, 5, 8, 6, 10, 5, 11, 6, 13, 7, 13, 7, 14, 5, 14, 7, 15, 8, 14, 4, 18, 8, 17, 7, 15, 5, 15, 11, 16, 8, 15, 7, 17, 12, 19, 10, 20, 10, 17, 10, 17, 13, 15, 11, 18, 8, 20, 10, 17, 9, 18, 11, 21, 11, 21, 7, 20, 11, 18, 11, 22, 9, 25, 11, 24, 13, 19, 14, 20, 11
(list; graph; listen)
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OFFSET
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1,8
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COMMENT
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Motivated by his conjecture related to A154257, on Dec 26, 2008 Zhi-Wei Sun conjectured that a(n)>0 for n=6,7,... 0n Jan 15 2009 Douglas McNeil verified this up to 10^12 and found no counterexamples. See the sequence A154536 for another conjecture of this sort. Sun also conjectured that any integer n>7 can be written as the sum of an odd prime, twice a positive Fibonacci number and the square of a positive Fibonacci number; this has been verified up to 2*10^8.
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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.
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.
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LINKS
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Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Douglas McNeil, Various and sundry (a report on Sun's conjectures)
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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FORMULA
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a(n)=|{<p,s,t>: p+F_s+2F_t=n with p an odd prime and s,t>1}|
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EXAMPLE
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For n=10 the a(10)=6 solutions are 3+F_4+2F_3, 3+F_5+2F_2, 3+F_2+2F_4, 5+F_2+2F_3, 5+F_4+2F_2, 7+F_2+2F_2.
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MATHEMATICA
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PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-2*Fibonacci[x]-Fibonacci[y]], 1, 0], {x, 2, 2*Log[2, Max[2, n/2]]}, {y, 2, 2*Log[2, Max[2, n-2*Fibonacci[x]]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 100000}]
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CROSSREFS
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Cf. A000040, A000045, A154257, A154258, A154263, A154285, A154290, A154417, A154536, A154404, A154940
Sequence in context: A011209 A071018 A144559 this_sequence A038572 A060992 A064455
Adjacent sequences: A155111 A155112 A155113 this_sequence A155115 A155116 A155117
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KEYWORD
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nice,nonn
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AUTHOR
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Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 20 2009
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