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%I #17 Sep 10 2018 14:03:46
%S 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,
%T 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,
%U 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
%N Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number.
%C 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.
%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="/A155114/b155114.txt">Table of n, a(n) for n = 1..100000</a>
%H D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;ab1b9553.0901">Various and sundry (a report on Sun's conjectures)</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 (II)</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/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>, Math. Comp. 78 (2009) 1853-1866. arXiv:math.NT/0702382.
%F a(n) = |{<p,s,t>: p+F_s+2F_t=n with p an odd prime and s,t>1}|.
%e 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.
%t 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}]
%Y Cf. A000040, A000045, A154257, A154258, A154263, A154285, A154290, A154417, A154536, A154404, A154940, A156695.
%K nice,nonn
%O 1,8
%A _Zhi-Wei Sun_, Jan 20 2009
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