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%I #10 Mar 02 2023 14:16:45
%S 0,0,0,1,2,2,2,2,3,1,3,4,2,2,3,1,2,3,3,2,4,1,2,5,2,3,3,1,3,2,1,3,4,1,
%T 2,5,2,2,6,3,2,3,3,2,4,1,3,3,2,1,3,2,2,6,3,4,7,4,5,7
%N Number of ways to write the (n+50)-th positive odd integer in the form p+2^x+51*2^y with p an odd prime and x,y positive integers.
%C Zhi-Wei Sun guessed that a(n)=0 if and only if n=1,2,3,127; in other words, except for 353, any odd integer greater than 105 can be written as the sum of an odd prime, a positive power of two and 51 times a positive power of two. D. S. McNeil has verified this for odd integers below 10^12. This is a part of the project for the form p+2^x+k*2^y with k=3,5,...,61 initiated by Zhi-Wei Sun in Jan. 2009.
%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="/A157372/b157372.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;20d0ba9.0902">A curious conjecture about p+2^x+11*2^y</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+51*2^y=2(n+50)-1 with p an odd prime and x,y positive integers}|
%e For n = 9 the a(9) = 3 solutions are: 2*59-1 = 7+2^3+51*2 = 11+2^2+51*2 = 13+2+51*2.
%t PQ[x_]:=x>1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2(n+50)-1-51*2^x-2^y],1,0], {x,1,Log[2,(2(n+50)-1)/51]},{y,1,Log[2,Max[2,2(n+50)-1-51*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]
%Y Cf. A000040, A000079, A157237, A157242, A155860, A155904, A156695, A154257, A154285, A155114, A154536.
%K nice,nonn
%O 1,5
%A _Zhi-Wei Sun_, Feb 28 2009
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