OFFSET
1,7
COMMENTS
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.
REFERENCES
R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
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.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
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
D. S. McNeil, Various and sundry (a report on Sun's conjectures)
Z.-W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075
FORMULA
a(n) = |{<p,x,y>: p+2^x+3*2^y = 2n-1 with p an odd prime and x,y positive integers}|.
EXAMPLE
For n=10 the a(10)=4 solutions are 19 = 3 + 2^2 + 3*2^2 = 5 + 2 + 3*2^2 = 5 + 2^3 + 3*2 = 11 + 2 + 3*2.
MATHEMATICA
PQ[x_]:=x>2&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-3*2^x-2^y], 1, 0], {x, 1, Log[2, (2n-1)/3]}, {y, 1, Log[2, Max[2, 2n-1-3*2^x]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
Zhi-Wei Sun, Jan 29 2009
STATUS
approved