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%I #14 Sep 11 2018 03:54:59
%S 0,0,0,0,0,0,0,0,0,0,0,1,1,0,2,1,0,2,3,1,1,3,1,1,4,2,3,2,1,3,3,2,3,5,
%T 1,2,5,2,4,5,1,4,3,1,4,7,1,5,7,2
%N Number of ways to write the n-th positive odd integer in the form p+2^x+7*2^y with p a prime congruent to 1 mod 6 and x,y positive integers.
%C On Feb 24 2009, _Zhi-Wei Sun_ conjectured that a(n)>0 for all n=18,19,...; in other words, any odd integer greater than 34 can be written as the sum of a prime congruent to 1 mod 6, a positive power of 2 and seven 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 R. 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 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
%H Zhi-Wei Sun, <a href="/A157218/b157218.txt">Table of n, a(n) for n=1..200000</a>
%H Z. W. Sun, <a href="http://arxiv.org/abs/0901.3075">Mixed sums of primes and other terms</a>, preprint, 2009. arXiv:0901.3075
%H 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 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;93b62faf.0901">A project for the form p+2^x+k*2^y with k=3,5,...,61</a>
%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6434d742.0812">A promising conjecture: n=p+F_s+F_t</a>
%H Z.-W. Sun and M.-H. Le, <a href="http://dx.doi.org/10.4064/aa99-2-5">Integers not of the form c*(2^a + 2^b) + p^{alpha}</a>, Acta Arith. 99(2001), 183-190.
%F a(n) = |{<p,x,y>: p+2^x+7*2^y=2n-1 with p a prime congruent to 1 mod 6 and x,y positive integers}|.
%e For n=19 the a(19)=3 solutions are 2*19 - 1 = 7 + 2 + 7*2^2 = 7 + 2^4 + 7*2 = 19 + 2^2 + 7*2.
%t PQ[x_]:=x>1&&Mod[x,6]==1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-7*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/7]},{y,1,Log[2,Max[2,2n-1-7*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]
%Y A000040, A000079, A155860, A155904, A156695, A154257, A154285, A155114, A154536, A154404, A154940.
%K nice,nonn
%O 1,15
%A _Zhi-Wei Sun_, Feb 25 2009
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