1,7

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 Greathouse continued the verfication 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 Erdos ever 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 positive powers of two.

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.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. http://arxiv.org/abs/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.

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

Douglas McNeil, Various and sundry (a report on Sun's conjectures)

a(n)=|{<p,x,y>: p+2^x+3*2^y=2n-1 with p an odd prime and x,y positive integers}|

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.

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}]

A000040, A000079, A154257, A154285, A155114, A154536, A154404, A154940

Sequence in context: A065648 A072813 A120887 this_sequence A030385 A031250 A031233

Adjacent sequences: A155857 A155858 A155859 this_sequence A155861 A155862 A155863

nice,nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 29 2009