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A154257 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. 20


%S 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,

%T 15,8,12,12,13,14,13,10,16,10,13,16,11,16,11,14,17,16,15,12,12,16,11,

%U 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

%N 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.

%C 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.

%C _Zhi-Wei Sun_ has offered a monetary reward for settling this conjecture.

%D R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

%D 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.

%H Zhi-Wei Sun, <a href="/A154257/b154257.txt">Table of n, a(n), n=1..50000.</a>

%H D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bda1acb4.0812">Sun's strong conjecture</a>

%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6434d742.0812">A promising conjecture: n=p+F_s+F_t</a>

%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;a9e706e.0812">A summary concerning my conjecture n=p+F_s+F_t (II)</a>

%H 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, <a href="http://dx.doi.org/10.1090/S0025-5718-09-02212-1">[DOI]</a>, <a href="http://arxiv.org/abs/math/0702382">arXiv:math.NT/0702382</a>

%e 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.

%t 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}]

%Y Cf. A000040, A000045, A156695. See A144559 for another version.

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Jan 05 2009

%E The new verification record is 10^14 (due to _D. S. McNeil_). - _Zhi-Wei Sun_, Jan 17 2009

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