A154285 Number of ordered triples satisfying p+L_s+L_t=n, where p is an odd prime, s and t are nonnegative and the Lucas number L_s or L_t is odd.

0, 0, 0, 0, 1, 2, 3, 6, 4, 8, 5, 8, 6, 8, 9, 12, 10, 12, 9, 10, 12, 14, 9, 14, 12, 14, 10, 14, 8, 10, 10, 16, 11, 16, 12, 18, 12, 16, 10, 12, 13, 16, 15, 16, 13, 14, 13, 16, 14, 18

Offset: 1

Author

Zhi-Wei Sun, Jan 06 2009

Keywords

• nonn

Comments

Zhi-Wei Sun conjectured that a(n)>0 for all n=5,6,...; in other words, any integer n>4 can be represented as the sum of an odd prime, an odd Lucas number and a Lucas number. This has been verified up to 1.5*10^8. Sun thought that the constant lim inf_n a(n)/log(n) is greater than 2 and smaller than 3. For k=2,3 Sun also conjectured that any integer n>4 can be written in the form p+L_s+(L_t)^k, where p is an odd prime and L_s or L_t is odd.

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

Examples

			For n=3 the a(7)=4 solutions are 3+L_1+L_2, 3+L_2+L_1, 5+L_1+L_1.
		

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

Crossrefs

Cf. A000032, A000040, A154257, A154258, A154263, A156695.

Programs

Mathematica

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n,2]==0||Mod[x,3]>0)&&PQ[n-(2*Fibonacci[x+1]-Fibonacci[x])-(2*Fibonacci[y+1]-Fibonacci[y])],1,0], {x,0,2*Log[2,n]},{y,0,2*Log[2,Max[1,n-(2*Fibonacci[x+1]-Fibonacci[x])]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]