%I #17 Sep 10 2018 13:49:49
%S 3,4,5,6,7,9,11,12,14,15,16,22,25,26,29,31,33,36,39,53,59,60,63,70,72,
%T 74,80,87,94,128,141,142,145,152,169,171,173,179,193,210,227,309,339,
%U 340,343,350,367,408,410,412
%N Positive integers that can be written as the sum of a positive Pell number and twice a positive Pell number.
%C On Jan 10 2009, _Zhi-Wei Sun_ conjectured that any integer greater than 5 can be expressed as the sum of an odd prime and a term in the above sequence; in other words, each n=6,7,... can be written in the form p+P_s+2*P_t with p an odd prime and s,t>0. This has been verified up to 5*10^13 by _D. S. McNeil_ (from London Univ.). Motivated by this conjecture, Qing-Hu Hou (from Nankai Univ.) observed and _Zhi-Wei Sun_ proved that each term a(n) in the above sequence can be uniquely written in the form P_s+2P_t with s,t>0. Sun noted that 2176 cannot be written as the sum of a prime and two Pell numbers; _D. S. McNeil_ found that 393185153350 cannot be written in the form p+P_s+3P_t and 872377759846 cannot be written in the form p+P_s+4P_t, where p is a prime and s and t are nonnegative.
%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.
%H Zhi-Wei SUN, <a href="/A154536/b154536.txt">Table of n, a(n), n=1..179.</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 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;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, 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., in press. <a href="http://arxiv.org/abs/math/0702382">arXiv:math.NT/0702382</a>
%e For n=12 the a(12)=22 solution is 22 = P_4 + 2*P_3.
%t P[n_]:=P[n]=2*P[n-1]+P[n-2] P[0]=0 P[1]=1 i:=0 Do[Do[If[n==2*P[x]+P[y],i=i+1;Print[i," ",n]], {x,1,Max[1,Log[2,n]]},{y,1,Log[2,n]+1}]; Continue,{n,1,100000}]
%Y Cf. A000129, A000040, A154257, A154285, A154364, A154417, A154421, A156695.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jan 11 2009
%E Mentioned McNeil's verification record for the representation n = p + P_s + 2P_t and his examples for n not of the form p + P_s + 3P_t and n not of the form p + P_s + 4P_t. - _Zhi-Wei Sun_, Jan 17 2009
%E _D. S. McNeil_ has verified the conjecture up to 5*10^13. - _Zhi-Wei Sun_, Jan 20 2009