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A154417
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Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and half of a positive Fibonacci number.
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4
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0, 0, 0, 0, 1, 1, 2, 2, 4, 3, 3, 4, 3, 4, 3, 4, 5, 2, 5, 5, 4, 6, 6, 4, 9, 5, 5, 6, 6, 5, 5, 6, 7, 3, 8, 6, 6, 7, 4, 5, 8, 5, 9, 4, 7, 6, 5, 7, 9, 5, 7, 4, 6, 6, 6, 7, 5, 4, 8, 3, 8, 8, 6, 6, 7, 7, 8, 6, 6, 6, 4, 6, 8, 3, 9, 8, 7, 10, 10, 8, 8, 8, 7, 6, 12, 7, 6, 10, 7, 7, 10, 10, 9, 5, 7, 11, 9, 10, 6, 6, 8
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OFFSET
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1,7
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COMMENTS
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On Jan 09, 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written in the form p+F_s+F_{3t}/2 with p an odd prime and s,t>0. Sun verified this up to 5*10^6 and Qing-Hu Hou continued the verification (on Sun's request) up to 3*10^8. Note that 932633 cannot be written as p+F_s+F_{3t}/2 with p a prime and (F_s or F_{3t}/2) odd. If we set u_0=0, u_1=1 and u_{n+1}=4u_n+u_{n-1} for n=1,2,3,..., then F_{3t}/2=u_t is at least 4^{t-1} for each t=1,2,3,.... In a recent paper K. J. Wu and Z. W. Sun constructed a residue class which contains no integers of the form p+F_{3t}/2 with p a prime and t nonnegative.
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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.
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), 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
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
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FORMULA
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a(n)=|{<p,s,t>: p+F_s+F_{3t}/2=n with p an odd prime, s>1 and t>0}|
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EXAMPLE
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For n=9 the a(9)=4 solutions are 3+F_5+F_3/2, 3+F_3+F_6/2, 5+F_4+F_3/2, 7+F_2+F_3/2.
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MATHEMATICA
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PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-Fibonacci[3x]/2-Fibonacci[y]], 1, 0], {x, 1, Log[2, n]+1}, {y, 2, 2*Log[2, Max[2, n-Fibonacci[3x]/2]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]
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CROSSREFS
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Cf. A000040, A000045, A154257, A154290, A156695.
Sequence in context: A002948 A117113 A162439 * A205563 A147594 A212652
Adjacent sequences: A154414 A154415 A154416 * A154418 A154419 A154420
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun, Jan 09 2009
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STATUS
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approved
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