%I #9 Sep 10 2018 14:47:12
%S 0,0,0,0,1,1,2,1,3,1,2,2,2,4,2,5,3,2,3,4,3,4,2,3,4,5,3,4,2,2,3,7,6,5,
%T 6,3,4,5,4,9,4,6,6,3,7,7,5,5,4,5
%N Number of triples <p, s,t> such that p+F_s+(F_t)^3=n, where p is an odd prime, s and t are greater than one and F_s or F_t is odd.
%C Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a cube of a positive Fibonacci number, with one of two Fibonacci numbers odd. He has verified this up to 3*10^7.
%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="/A154263/b154263.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</a>
%H K. J. Wu and Z. W. Sun, <a href="http://arxiv.org/abs/math.NT/0702382">Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2</a>, Math. Comp., in press. arXiv:math.NT/0702382
%e For n=14 the a(14)=4 solutions are 3+F_4+(F_3)^3, 5+F_2+(F_3)^3, 5+F_6+(F_2)^3, 11+F_3+(F_2)^3
%t PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n,2]==0||Mod[x,3]>0)&&PQ[n-(Fibonacci[x])^3-Fibonacci[y]],1,0], {x,2,2*Log[2,n^(1/3)+1]},{y,2,2*Log[2,Max[2,n-(Fibonacci[x])^3]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]
%Y Cf. A154257, A154258, A000040, A000045
%K nonn
%O 1,7
%A _Zhi-Wei Sun_, Jan 06 2009