A154417 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and half of a positive Fibonacci number.

0, 0, 0, 0, 1, 1, 2, 2, 4, 3, 3, 4, 3, 4, 3, 4, 5, 2, 5, 5, 4, 6, 6, 4, 9, 5, 5, 6, 6, 5, 5, 6, 7, 3, 8, 6, 6, 7, 4, 5, 8, 5, 9, 4, 7, 6, 5, 7, 9, 5, 7, 4, 6, 6, 6, 7, 5, 4, 8, 3, 8, 8, 6, 6, 7, 7, 8, 6, 6, 6, 4, 6, 8, 3, 9, 8, 7, 10, 10, 8, 8, 8, 7, 6, 12, 7, 6, 10, 7, 7, 10, 10, 9, 5, 7, 11, 9, 10, 6, 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 every n=5,6,...; in other words, any integer n>4 can be written in the form p+F_s+F_{3t}/2 with p an odd prime and s,t>0. Sun verified this up to 5*10^6 and Qing-Hu Hou continued the verification (on Sun's request) up to 3*10^8. Note that 932633 cannot be written as p+F_s+F_{3t}/2 with p a prime and (F_s or F_{3t}/2) odd. If we set u_0=0, u_1=1 and u_{n+1}=4u_n+u_{n-1} for n=1,2,3,..., then F_{3t}/2=u_t is at least 4^{t-1} for each t=1,2,3,.... In a recent paper K. J. Wu and Z. W. Sun constructed a residue class which contains no integers of the form p+F_{3t}/2 with p a prime and t nonnegative.

Examples

			For n=9 the a(9)=4 solutions are 3 + F_5 + F_3/2, 3 + F_3 + F_6/2, 5 + F_4 + F_3/2, 7 + F_2 + F_3/2.
		

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
• 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
• Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

Crossrefs

Cf. A000040, A000045, A154257, A154290, A156695.

Programs

Mathematica

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

Formula

a(n) = |{: p+F_s+F_{3t}/2=n with p an odd prime, s>1 and t>0}|.