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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 1 mod 6 and x,y positive integers.
3

%I #8 Sep 11 2018 03:55:59

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,2,1,0,2,2,0,1,1,1,2,2,2,4,1,2,5,

%T 2,1,3,1,1,2,1,3,3,1,3,5,2,2,5,4,0,5,4,2,4,3,3,4,3,3

%N 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 1 mod 6 and x,y positive integers.

%C 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.

%D R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

%D 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.

%H Zhi-Wei Sun, <a href="/A157237/b157237.txt">Table of n, a(n) for n=1..200000</a>

%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, <a href="http://arxiv.org/abs/0901.3075">Mixed sums of primes and other terms</a>, preprint, 2009. arXiv:0901.3075

%F 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}|

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

%t 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}]

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

%K nonn

%O 1,19

%A _Zhi-Wei Sun_, Feb 25 2009