%I A155904
%S A155904 0,0,0,0,0,0,0,1,2,2,2,2,4,3,5,6,4,5,4,4,6,5,6,7,7,5,7,11,5,10,8,5,10,
               7,
%T A155904 5,8,8,7,6,10,6,8,13,9,12,10,8,14,10,7,13,12,7,10,10,9,10,17,8,11,11,9,
%U A155904 16,12,7,13,8,10,7,8,10,11,14,5,14,14,10,17,12,7,11,12,10,12,10,12,13,
               17
%N A155904 Number of ways to write 2n-1 as p+2^x+5*2^y with p an odd prime and x,
               y positive integers.
%C A155904 On Jan 21, 2009 Zhi-Wei Sun conjectured that a(n)>0 for n=8,9,...; in 
               other words, any odd integer m>=15 can be written as the sum of an 
               odd prime, a positive power of 2 and five times a positive power 
               of 2. Sun has verified this for odd integers m<10^8. As 5*2^y=2^y+2^{y+2}, 
               the conjecture implies that each odd integer m>8 can be written as 
               the sum of an odd prime and three positive powers of two. [It is 
               known that there are infinitely many positive odd integers not of 
               the form p+2^x+2^y (R. Crocker, 1971).] Sun also conjectured that 
               there are infinitely many positive integers n with a(n)=a(n+1); here 
               is the list of such positive integers n: 1, 2, 3, 4, 5, 6, 9, 10, 
               11, 19, 24, 36, 54, 60, 75, 90, 98, 101, 105, 135, 153, 173, ...
%D A155904 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 
               36(1971), 103-107.
%D A155904 Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. http:/
               /arxiv.org/abs/0901.3075
%D A155904 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 A155904 Zhi-Wei Sun, <a href="b155904.txt">Table of n, a(n) for n = 1..50000</
               a>
%H A155904 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 A155904 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>
%H A155904 Douglas McNeil, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0901&L=nmbrthry&T=0&P=840">
               Various and sundry (a report on Sun's conjectures)</a>
%F A155904 a(n)=|{<p,x,y>: p+2^x+5*2^y=2n-1 with p an odd prime and x,y positive 
               integers}|
%e A155904 For n=15 the a(15)=5 solutions are 29=17+2+5*2=11+2^3+5*2=3+2^4+5*2=7+2+5*2^2=5+2^2+5*2^2.
%t A155904 PQ[x_]:=x>2&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-5*2^x-2^y],1,0], {x,1,Log[2,
               (2n-1)/5]},{y,1,Log[2,2n-1-5*2^x]}] Do[Print[n," ",RN[n]];Continue,
               {n,1,50000}]
%Y A155904 Cf. A000040, A000079, A155860, A154257, A154285, A155114, A154536, A154404, 
               A154940
%Y A155904 Sequence in context: A097576 A029250 A110884 this_sequence A125913 A122386 
               A051464
%Y A155904 Adjacent sequences: A155901 A155902 A155903 this_sequence A155905 A155906 
               A155907
%K A155904 nice,nonn
%O A155904 1,9
%A A155904 Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Jan 30 2009