cp's OEIS Frontend

A154258 Number of triples such that p+F_s+(F_t)^2=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.

%I A154258 #9 Sep 10 2018 14:40:13
%S A154258 0,0,0,0,1,1,2,2,3,3,2,4,3,5,4,5,6,3,5,6,5,7,4,5,7,4,6,7,6,6,6,5,11,6,
%T A154258 8,6,6,7,6,9,9,4,9,5,9,10,6,8,8,7
%N A154258 Number of triples <p,s,t> such that p+F_s+(F_t)^2=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 A154258 Zhi-Wei Sun conjectured 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, a positive Fibonacci number and a square of a positive Fibonacci number, with one of the two Fibonacci numbers odd). He has verified this for n up to 3*10^7.
%C A154258 _Zhi-Wei Sun_ has offered a monetary reward for settling this conjecture.
%D A154258 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
%D A154258 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 A154258 Zhi-Wei SUN, <a href="/A154258/b154258.txt">Table of n, a(n), n=1..50000.</a>
%H A154258 D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bda1acb4.0812">Sun's strong conjecture</a>
%H A154258 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6434d742.0812">A summary concerning my conjecture n=p+F_s+F_t</a>
%H A154258 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 A154258 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 A154258 For n=10 the a(10)=3 solutions are 3+F_4+(F_3)^2, 5+F_2+(F_3)^2, 7+F_3+(F_2)^2.
%t A154258 PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n,2]==0||Mod[x,3]>0)&&PQ[n-(Fibonacci[x])^2-Fibonacci[y]],1,0], {x,2,2*Log[2,Sqrt[n]+1]},{y,2,2*Log[2,Max[2,n-(Fibonacci[x])^2]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]
%Y A154258 Cf. A000040, A000045, A154257
%K A154258 nonn
%O A154258 1,7
%A A154258 _Zhi-Wei Sun_, Jan 05 2009