1,19

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<16 or n=18, 21, 24, 51, 84, 1011, 59586; in other words, except for 35, 41, 47, 101, 167, 2021, 119171, any odd integer greater than 30 can be written as the sum of a prime congruent to 1 mod 6, a positive power of 2 and eleven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7, and Qing-Hu Hou continued the verification for odd integers below 1.5*10^8 (on Sun's request). Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.

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 promising conjecture: n=p+F_s+F_t

a(n)=|{<p,x,y>: p+2^x+11*2^y=2n-1 with p a prime congruent to 1 mod 6 and x,y positive integers}|

For n=19 the a(19)=2 solutions are 2*19-1=7+2^3+2*11=13+2+2*11.

PQ[x_]:=x>1&&Mod[x, 6]==1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-11*2^x-2^y], 1, 0], {x, 1, Log[2, (2n-1)/11]}, {y, 1, Log[2, Max[2, 2n-1-11*2^x]]}] Do[Print[n, " ", RN[n]], {n, 1, 200000}]

A000040, A000079, A157218, A157225, A155860, A155904, A156695, A154257, A154285, A155114, A154536

Sequence in context: A134663 A000925 A003985 this_sequence A065676 A146973 A003263

Adjacent sequences: A157234 A157235 A157236 this_sequence A157238 A157239 A157240

nonn

Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Feb 25 2009