1,5

Zhi-Wei Sun guessed that a(n)=0 if and only if n=1,2,3,127; in other words, except for 353, any odd integer greater than 105 can be written as the sum of an odd prime, a positive power of two and 51 times a positive power of two. Douglas McNeil has verified this for odd integers below 10^12. This is a part of the project for the form p+2^x+k*2^y with k=3,5,...,61 initiated by Zhi-Wei Sun in Jan. 2009.

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.

Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. http://arxiv.org/abs/0901.3075

Zhi-Wei Sun, Table of n, a(n) for n=1..200000

Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.

Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61

Zhi-Wei Sun, A curious conjecture about p+2^x+11*2^y

a(n)=|{<p,x,y>: p+2^x+51*2^y=2(n+50)-1 with p an odd prime and x,y positive integers}|

For n=9 the a(9)=3 solutions are 2*59-1=7+2^3+51*2=11+2^2+51*2=13+2+51*2

PQ[x_]:=x>1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2(n+50)-1-51*2^x-2^y], 1, 0], {x, 1, Log[2, (2(n+50)-1)/51]}, {y, 1, Log[2, Max[2, 2(n+50)-1-51*2^x]]}] Do[Print[n, " ", RN[n]], {n, 1, 200000}]

A000040, A000079, A157237, A157242, A155860, A155904, A156695, A154257, A154285, A155114, A154536

Sequence in context: A063473 A096859 A005086 this_sequence A020649 A067131 A094915

Adjacent sequences: A157369 A157370 A157371 this_sequence A157373 A157374 A157375

nice,nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Feb 28 2009