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%I #10 Apr 17 2022 22:26:35
%S 0,0,0,0,0,0,0,0,0,0,1,1,0,2,1,0,2,3,1,2,4,1,2,4,2,2,3,2,2,4,2,4,4,1,
%T 5,5,2,5,7,1,3,7,2,4,8,2,4,3,2,4
%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 5 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<11 or n=13,16,992; in other words, except for 25, 31, 1983, any odd integer greater than 20 can be written as the sum of a prime congruent to 5 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 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="/A157225/b157225.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+7*2^y=2n-1 with p a prime congruent to 5 mod 6 and x,y positive integers}|
%e For n=18 the a(18)=3 solutions are 2*18-1=5+2+7*2^2=5+2^4+7*2=17+2^2+7*2.
%t PQ[x_]:=x>1&&Mod[x,6]==5&&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 Cf. A000040, A000079, A157218, A155860, A155904, A156695, A154257, A154285, A155114, A154536, A154404, A154940.
%K nice,nonn
%O 1,14
%A _Zhi-Wei Sun_, Feb 25 2009
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