1,1

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 Douglas 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. 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.

Zhi-Wei Sun (zwsun(AT)nju.edu.cn) has offered a monetary reward for settling this conjecture.

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

T. Tao, A remark on primality testing and decimal expansions, J. Austral. Math. Soc., in press. arXiv:0802.3361

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. arXiv:math.NT/0702382

Zhi-Wei SUN, Table of n, a(n), n=1..179.

Douglas McNeil, Sun's strong conjecture

Douglas McNeil, Various and sundry: a report on Sun's conjectures

Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t

For n=12 the a(12)=22 solution is 22=P_4+2*P_3.

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}]

Cf. A000129, A000040, A154257, A154285, A154364, A154417, A154421.

Sequence in context: A048869 A039051 A047564 this_sequence A091815 A081692 A161346

Adjacent sequences: A154533 A154534 A154535 this_sequence A154537 A154538 A154539

nice,nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 11 2009

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 (zwsun(AT)nju.edu.cn), Jan 17 2009

D. McNeil has verified the conjecture up to 5*10^13. - Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 20 2009