A154263 Number of triples 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.

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, 6, 3, 4, 5, 4, 9, 4, 6, 6, 3, 7, 7, 5, 5, 4, 5

Offset: 1

Author

Zhi-Wei Sun, Jan 06 2009

Keywords

• nonn

Comments

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.

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

Examples

			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
		

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 promising conjecture: n=p+F_s+F_t
• Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t
• 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. A154257, A154258, A000040, A000045

Programs

Mathematica

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}]