cp's OEIS Frontend

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.

%I A154417 #19 Sep 10 2018 14:41:44
%S A154417 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,
%T A154417 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,
%U A154417 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
%N 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.
%C A154417 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.
%D A154417 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
%H A154417 Zhi-Wei SUN, <a href="/A154417/b154417.txt">Table of n, a(n), n=1..50000.</a>
%H A154417 D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bda1acb4.0812">Sun's strong conjecture</a>
%H A154417 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6434d742.0812">A promising conjecture: n=p+F_s+F_t</a>
%H A154417 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;a9e706e.0812">A summary concerning my conjecture n=p+F_s+F_t</a>
%H A154417 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, <a href="http://dx.doi.org/10.1090/S0025-5718-09-02212-1">[DOI]</a>, <a href="http://arxiv.org/abs/math/0702382">arXiv:math.NT/0702382</a>
%H A154417 Terence Tao, <a href="http://arxiv.org/abs/0802.3361">A remark on primality testing and decimal expansions</a>, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
%F A154417 a(n) = |{<p,s,t>: p+F_s+F_{3t}/2=n with p an odd prime, s>1 and t>0}|.
%e A154417 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.
%t A154417 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}]
%Y A154417 Cf. A000040, A000045, A154257, A154290, A156695.
%K A154417 nonn
%O A154417 1,7
%A A154417 _Zhi-Wei Sun_, Jan 09 2009