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A related problem is to determine the index of the last occurrence of n in A154404. Among the first 10^6 terms in A154404, the values 0, 1, 2 and 3 last occur at indices 4, 5, 6 and 8, respectively, but all values larger than 3 that occur at all (4 through 56 and 58 through 61) do so at least once beyond the 500000th term.

The value 4, after its initial occurrence in A154404 at n=12, does not reoccur until n=666393. (The 4 ways to reach 666393 as a sum of an odd prime, a positive Fibonacci number and a Catalan number are 605023+2584+58786, 606997+610+58786, 607573+34+58786 and 648677+17711+5.)

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 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, ...

On Feb 24 2009, Zhi-Wei Sun conjectured that a(n)>0 for all n=18,19,...; in other words, any odd integer greater than 34 can be written as the sum of a prime congruent to 1 mod 6, a positive power of 2 and seven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7, and Qing-Hu Hou continued the verification for odd integers below 1.5*10^8 (on Sun's request). Compare the conjecture with R. Crocker's result that there are infinitely many positive odd integers not of the form p + 2^x + 2^y with p an odd prime and x,y positive integers.

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<11 or n=13,16,992; in other words, except for 25, 31, 1983, any odd integer greater than 20 can be written as the sum of a prime congruent to 5 mod 6, a positive power of 2 and seven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7, and Qing-Hu Hou continued the verification for odd integers below 1.5*10^8 (on Sun's request). Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.