A154290 - OEIS

%I #11 Sep 10 2018 14:06:44

%S 0,0,0,0,1,2,3,5,5,7,6,8,6,8,8,10,9,9,11,11,10,14,10,11,11,15,13,14,

%T 10,10,11,12,12,14,15,14,13,14,12,13,11,16,13,15,15,16,13,17,12,17

%N Number of ordered triples <p,s,t> satisfying p+F_s+L_t = n, where p is an odd prime, s >= 2 and F_s or L_t is odd.

%C Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a Lucas number, with the Fibonacci number or the Lucas number odd. Moreover, Sun conjectured that lim inf_n a(n)/log(n) is greater than 3 and smaller than 4.

%D R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

%H Zhi-Wei SUN, <a href="/A154290/b154290.txt">Table of n, a(n), n=1..50000.</a>

%H D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bda1acb4.0812">Sun's strong conjecture</a>

%H 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 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 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.

%H K. J. Wu and Z. W. Sun, <a href="http://arxiv.org/abs/math.NT/0702382">Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2</a>, arXiv:math.NT/0702382, Math. Comp. 78 (2009) 1853-1868.

%e For n=10 the a(10)=7 solutions are 3+F_4+L_3, 3+F_5+L_0, 5+F_2+L_3, 5+F_3+L_2, 5+F_4+L_0, 7+F_2+L_0, 7+F_3+L_1.

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

%Y Cf. A000040, A000045, A000032, A154257, A154285

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Jan 06 2009, Jan 07 2008