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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.

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

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 Zhi-Wei Sun's conjecture that each integer n>4 can be expressed as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number (cf. A154257), during their visit to Nanjing Univ. Qing-Hu Hou (Nankai Univ.) and Jiang Zeng (Univ. of Lyon-I) conjectured on Jan 09 2009 that a(n)>0 for every n=5,6,.... and verified this up to 5*10^8. D. S. McNeil has verified the conjecture up to 5*10^13 and Hou and Zeng have offered prizes for settling their conjecture (see Sun 2009).

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.

Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a Lucas number, with the Fibonacci number or the Lucas number odd. Moreover, Sun conjectured that lim inf_n a(n)/log(n) is greater than 3 and smaller than 4.

On Jan 21 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=6,7,...; in other words, any odd integer m>10 can be written as the sum of an odd prime, a positive power of 2 and three times a positive power of 2. Sun verified this for odd integers m<10^7. On Sun's request, Qing-Hu Hou and Charles R Greathouse IV continued the verification for odd integers below 2*10^8 and 10^10 respectively and found no counterexamples.

As 3*2^y = 2^y + 2^{y+1}, Sun's conjecture implies that each odd integer m>8 can be written as the sum of an odd prime and three positive powers of two. Note that Paul Erdős asked whether there is a positive integer r such that every odd integer m>3 can be written as the sum of a prime and at most r powers of 2.

Zhi-Wei Sun also raised the following problem: For k=3,5,...,61 determine whether any odd integer m>2k+3 can be written in the form p + 2^x + k*2^y with p an odd prime and x,y positive integers. Sun observed that 353 is not of the form p + 2^x + 51*2^y and Qing-Hu Hou continued the search for m<2.5*10^7 and found that 22537515 is not of the form p + 2^x + 47*2^y. For k=3,5,...,45,49,53,55,...,61, Sun has checked odd integers below 10^8 and found no odd integer m>2k-3 not of the form p + 2^x + k*2^y.

On Jan 16 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=5,6,... and verified this up to 5*10^6. (Sun also thought that lim inf_n a(n)/log(n) is a positive constant.) D. S. McNeil continued the verification up to 10^13 and found no counterexamples. The conjecture is similar to a conjecture of Qing-Hu Hou and Jiang Zeng related to the sequence A154404; both conjectures were motivated by Sun's recent conjecture on sums of primes and Fibonacci numbers (cf. A154257).

On Jan 21 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=8,9,...; in other words, any odd integer m>=15 can be written as the sum of an odd prime, a positive power of 2 and five times a positive power of 2. Sun has verified this for odd integers m<10^8. As 5*2^y=2^y+2^{y+2}, the conjecture implies that each odd integer m>8 can be written as the sum of an odd prime and three positive powers of two. [It is known that there are infinitely many positive odd integers not of the form p+2^x+2^y (R. Crocker, 1971).] Sun also conjectured that there are infinitely many positive integers n with a(n)=a(n+1); here is the list of such positive integers n: 1, 2, 3, 4, 5, 6, 9, 10, 11, 19, 24, 36, 54, 60, 75, 90, 98, 101, 105, 135, 153, 173, ...

Zhi-Wei Sun conjectured that a(n)>0 for all n=5,6,... (i.e., any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a square of a positive Fibonacci number, with one of the two Fibonacci numbers odd). He has verified this for n up to 3*10^7.

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

On Jan 09 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written in the form p+F_s+F_{3t}/2 with p an odd prime and s,t>0. Sun verified this up to 5*10^6 and Qing-Hu Hou continued the verification (on Sun's request) up to 3*10^8. Note that 932633 cannot be written as p+F_s+F_{3t}/2 with p a prime and (F_s or F_{3t}/2) odd. If we set u_0=0, u_1=1 and u_{n+1}=4u_n+u_{n-1} for n=1,2,3,..., then F_{3t}/2=u_t is at least 4^{t-1} for each t=1,2,3,.... In a recent paper K. J. Wu and Z. W. Sun constructed a residue class which contains no integers of the form p+F_{3t}/2 with p a prime and t nonnegative.