

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.


4



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
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OFFSET

1,7


COMMENTS

On Jan 09 2009, ZhiWei 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 QingHu 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_{n1} for n=1,2,3,..., then F_{3t}/2=u_t is at least 4^{t1} 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.


REFERENCES

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103107.


LINKS

K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m2^n and x^2F_{3n}/2, Math. Comp. 78 (2009) 1853, [DOI], arXiv:math.NT/0702382


FORMULA

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


EXAMPLE

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.


MATHEMATICA

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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



