1,9

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, ...

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+5*2^y=2n-1 with p an odd prime and x,y positive integers}|

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.

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}]

Cf. A000040, A000079, A155860, A154257, A154285, A155114, A154536, A154404, A154940

Adjacent sequences: A155901 A155902 A155903 this_sequence A155905 A155906 A155907

Sequence in context: A097576 A029250 A110884 this_sequence A125913 A122386 A051464

nice,nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 30 2009