%I A157242
%S A157242 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,2,1,0,2,2,0,2,2,1,3,2,2,3,1,2,4,2,2,
%T A157242 5,1,2,5,2,2,4,2,2,3,2,3,4,4,3,6,2,3,6,5,1,7,4,2,6
%N A157242 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y 
               with p a prime congruent to 5 mod 6 and x,y positive integers.
%C A157242 On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if 
               n<15 or n=17, 20, 23, 86, 124; in other words, except for 33, 39, 
               45, 171 and 247, any odd integer greater than 28 can be written as 
               the sum of a prime p=5 (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. Knowing the conjecture from Sun, Qing-Hu Hou 
               and Douglas McNeil have continued the verification for odd integers 
               below 1.5*10^8 and 10^12 respectively, and they have found no counterexample. 
               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.
%D A157242 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 
               36(1971), 103-107.
%D A157242 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.
%D A157242 Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. http:/
               /arxiv.org/abs/0901.3075
%H A157242 Zhi-Wei Sun, <a href="b157242.txt">Table of n, a(n) for n=1..200000</
               a>
%H A157242 Zhi-Wei Sun, A webpage: <a href="http://math.nju.edu.cn/~zwsun/MSPT.htm">
               Mixed Sums of Primes and Other Terms</a>, 2009.
%H A157242 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0901&L=nmbrthry&T=0&P=1886">
               A project for the form p+2^x+k*2^y with k=3,5,...,61</a>
%H A157242 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0812&L=nmbrthry&T=0&P=2140">
               A promising conjecture: n=p+F_s+F_t</a>
%F A157242 a(n)=|{<p,x,y>: p+2^x+11*2^y=2n-1 with p a prime congruent to 5 mod 6 
               and x,y positive integers}|
%e A157242 For n=18 the a(18)=2 solutions are 2*18-1=5+2^3+2*11=11+2+2*11.
%t A157242 PQ[x_]:=x>1&&Mod[x,6]==5&&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}]
%Y A157242 A000040, A000079, A157237, A155860, A155904, A156695, A154257, A154285, 
               A155114, A154536
%Y A157242 Sequence in context: A065676 A146973 A003263 this_sequence A135211 A029294 
               A065434
%Y A157242 Adjacent sequences: A157239 A157240 A157241 this_sequence A157243 A157244 
               A157245
%K A157242 nice,nonn
%O A157242 1,18
%A A157242 Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Feb 25 2009