A154290 Number of ordered triples satisfying p+F_s+L_t = n, where p is an odd prime, s >= 2 and F_s or L_t is odd.

0, 0, 0, 0, 1, 2, 3, 5, 5, 7, 6, 8, 6, 8, 8, 10, 9, 9, 11, 11, 10, 14, 10, 11, 11, 15, 13, 14, 10, 10, 11, 12, 12, 14, 15, 14, 13, 14, 12, 13, 11, 16, 13, 15, 15, 16, 13, 17, 12, 17

Offset: 1

Author

Zhi-Wei Sun, Jan 06 2009, Jan 07 2008

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 Lucas number, with the Fibonacci number or the Lucas number odd. Moreover, Sun conjectured that lim inf_n a(n)/log(n) is greater than 3 and smaller than 4.

Examples

			For n=10 the a(10)=7 solutions are 3+F_4+L_3, 3+F_5+L_0, 5+F_2+L_3, 5+F_3+L_2, 5+F_4+L_0, 7+F_2+L_0, 7+F_3+L_1.
		

References

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

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
• Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
• 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, arXiv:math.NT/0702382, Math. Comp. 78 (2009) 1853-1868.

Crossrefs

Cf. A000040, A000045, A000032, A154257, A154285

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])-Fibonacci[y]],1,0], {x,0,2*Log[2,n]},{y,2,2*Log[2,Max[2,n-(2*Fibonacci[x+1]-Fibonacci[x])]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]