OFFSET
1,9
COMMENTS
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, ...
REFERENCES
R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
D. S. McNeil, Various and sundry (a report on Sun's conjectures)
Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075
Z.-W. Sun and M.-H. Le, Integers not of the form c*(2^a + 2^b) + p^{alpha}, Acta Arith. 99(2001), 183-190.
FORMULA
a(n) = |{<p,x,y>: p+2^x+5*2^y=2n-1 with p an odd prime and x,y positive integers}|.
EXAMPLE
For n=15 the a(15)=5 solutions are 29 = 17 + 2 + 5*2 = 11 + 2^3 + 5*2 = 3 + 2^4 + 5*2 = 7 + 2 + 5*2^2 = 5 + 2^2 + 5*2^2.
MATHEMATICA
PQ[x_]:=x>2&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-5*2^x-2^y], 1, 0], {x, 1, Log[2, (2n-1)/5]}, {y, 1, Log[2, 2n-1-5*2^x]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
Zhi-Wei Sun, Jan 30 2009
STATUS
approved