A154257 Number of triples 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.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 05 2009

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

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 (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. 78 (2009) 1853, [DOI], arXiv:math.NT/0702382

Crossrefs

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

Programs

Mathematica

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}]

Extensions

The new verification record is 10^14 (due to D. S. McNeil). - Zhi-Wei Sun, Jan 17 2009