cp's OEIS Frontend

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.

Zhi-Wei Sun has offered a monetary reward for settling this conjecture.

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.

On Jan 09 2009, Zhi-Wei Sun conjectured that a(n)>0 for all n=6,7,.... ; in other words, any integer n>5 can be written in the form p+F_s+L_{3t} with p an odd prime, s positive and t nonnegative. [Compare this with the conjecture related to the sequence A154290.] Sun verified the above conjecture up to 5*10^6 and Qing-Hu Hou continued the verification up to 2*10^8. If we set v_0=2, v_1=4 and v_{n+1}=4v_n+v_{n-1} for n=1,2,3,..., then L_{3t}=v_t is at least 4^t for every t=0,1,2,.... On Jan 17 2009, D. S. McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.