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.

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, 8, 6, 6, 7, 6, 9, 9, 4, 9, 5, 9, 10, 6, 8, 8, 7

Offset: 1

Author

Zhi-Wei Sun, Jan 05 2009

Keywords

• nonn

Comments

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.

Zhi-Wei Sun has offered a monetary reward for settling this conjecture.

Examples

			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.
		

References

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

Links

• Zhi-Wei SUN, Table of n, a(n), n=1..50000.
• D. S. McNeil, Sun's strong conjecture
• Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t
• Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t (II)
• 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

Crossrefs

Cf. A000040, A000045, A154257

Programs

Mathematica

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}]