A154421 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.

0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 2, 5, 2, 4, 3, 4, 4, 5, 2, 6, 2, 7, 5, 7, 3, 9, 3, 9, 4, 7, 3, 6, 4, 9, 3, 10, 3, 8, 4, 6, 5, 8, 6, 8, 3, 9, 4, 8, 6, 8

Offset: 1

Author

Zhi-Wei Sun, Jan 09 2009

Keywords

• nonn

Comments

On Jan 09 2009, Zhi-Wei Sun conjectured that a(n)>0 for all n=6,7,.... ; in other words, any integer n>5 can be written in the form p+F_s+L_{3t} with p an odd prime, s positive and t nonnegative. [Compare this with the conjecture related to the sequence A154290.] Sun verified the above conjecture up to 5*10^6 and Qing-Hu Hou continued the verification up to 2*10^8. If we set v_0=2, v_1=4 and v_{n+1}=4v_n+v_{n-1} for n=1,2,3,..., then L_{3t}=v_t is at least 4^t for every t=0,1,2,.... On Jan 17 2009, D. S. McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.

Examples

			For n=8 the a(8)=3 solutions are 3 + F_4 + L_0, 3 + F_2 + L_3, 5 + F_2 + L_0.
		

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

Crossrefs

Cf. A000040, A000045, A000032, A154257, A154285, A154290, A154364, A154417, A156695.

Programs

Mathematica

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

Formula

a(n) = |{: p+F_s+L_{3t}=n with p an odd prime, s>1 and t nonnegative}|.

Extensions

McNeil's counterexample added by Zhi-Wei Sun, Jan 20 2009