A155114 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number.

0, 0, 0, 0, 0, 1, 1, 3, 2, 6, 3, 7, 3, 8, 5, 8, 6, 10, 5, 11, 6, 13, 7, 13, 7, 14, 5, 14, 7, 15, 8, 14, 4, 18, 8, 17, 7, 15, 5, 15, 11, 16, 8, 15, 7, 17, 12, 19, 10, 20, 10, 17, 10, 17, 13, 15, 11, 18, 8, 20, 10, 17, 9, 18, 11, 21, 11, 21, 7, 20, 11, 18, 11, 22, 9, 25, 11, 24, 13, 19, 14, 20, 11

Offset: 1

Author

Zhi-Wei Sun, Jan 20 2009

Keywords

• nice
• nonn

Comments

Motivated by his conjecture related to A154257, on Dec 26 2008, Zhi-Wei Sun conjectured that a(n)>0 for n=6,7,... On Jan 15 2009, D. S. McNeil verified this up to 10^12 and found no counterexamples. See the sequence A154536 for another conjecture of this sort. Sun also conjectured that any integer n>7 can be written as the sum of an odd prime, twice a positive Fibonacci number and the square of a positive Fibonacci number; this has been verified up to 2*10^8.

Examples

			For n=10 the a(10)=6 solutions are 3 + F_4 + 2F_3, 3 + F_5 + 2F_2, 3 + F_2 + 2F_4, 5 + F_2 + 2F_3, 5 + F_4 + 2F_2, 7 + F_2 + 2F_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) for n = 1..100000
• D. S. McNeil, Various and sundry (a report on Sun's conjectures)
• 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)
• 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) 1853-1866. arXiv:math.NT/0702382.

Crossrefs

Cf. A000040, A000045, A154257, A154258, A154263, A154285, A154290, A154417, A154536, A154404, A154940, A156695.

Programs

Mathematica

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-2*Fibonacci[x]-Fibonacci[y]],1,0], {x,2,2*Log[2,Max[2,n/2]]},{y,2,2*Log[2,Max[2,n-2*Fibonacci[x]]]}] Do[Print[n," ",RN[n]];Continue,{n,1,100000}]

Formula

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