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A154285 Number of ordered triples <p,s,t> satisfying p+L_s+L_t=n, where p is an odd prime, s and t are nonnegative and the Lucas number L_s or L_t is odd. 15


%S 0,0,0,0,1,2,3,6,4,8,5,8,6,8,9,12,10,12,9,10,12,14,9,14,12,14,10,14,8,

%T 10,10,16,11,16,12,18,12,16,10,12,13,16,15,16,13,14,13,16,14,18

%N Number of ordered triples <p,s,t> satisfying p+L_s+L_t=n, where p is an odd prime, s and t are nonnegative and the Lucas number L_s or L_t is odd.

%C Zhi-Wei Sun conjectured that a(n)>0 for all n=5,6,...; in other words, any integer n>4 can be represented as the sum of an odd prime, an odd Lucas number and a Lucas number. This has been verified up to 1.5*10^8. Sun thought that the constant lim inf_n a(n)/log(n) is greater than 2 and smaller than 3. For k=2,3 Sun also conjectured that any integer n>4 can be written in the form p+L_s+(L_t)^k, where p is an odd prime and L_s or L_t is odd.

%C _Zhi-Wei Sun_ has offered a monetary reward for settling this conjecture.

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

%D Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

%H Zhi-Wei SUN, <a href="/A154285/b154285.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 summary concerning my 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 (II)</a>

%H 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, <a href="http://dx.doi.org/10.1090/S0025-5718-09-02212-1">Math. Comp. 78 (2009) 1853</a>, <a href="http://arxiv.org/abs/math/0702382">arXiv:math.NT/0702382</a>

%e For n=3 the a(7)=4 solutions are 3+L_1+L_2, 3+L_2+L_1, 5+L_1+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])-(2*Fibonacci[y+1]-Fibonacci[y])],1,0], {x,0,2*Log[2,n]},{y,0,2*Log[2,Max[1,n-(2*Fibonacci[x+1]-Fibonacci[x])]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]

%Y Cf. A000032, A000040, A154257, A154258, A154263, A156695.

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Jan 06 2009

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Last modified April 7 11:37 EDT 2020. Contains 333304 sequences. (Running on oeis4.)