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%I #15 Sep 10 2018 14:17:43
%S 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,
%T 7,5,8,8,7,6,10,6,8,13,9,12,10,8,14,10,7,13,12,7,10,10,9,10,17,8,11,
%U 11,9,16,12,7,13,8,10,7,8,10,11,14,5,14,14,10,17,12,7,11,12,10,12,10,12,13,17
%N 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 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 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="/A155904/b155904.txt">Table of n, a(n) for n = 1..50000</a>
%H D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;ab1b9553.0901">Various and sundry (a report on Sun's conjectures)</a>
%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
%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+5*2^y=2n-1 with p an odd prime and x,y positive integers}|.
%e 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 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 Cf. A000040, A000079, A155860, A154257, A154285, A155114, A154536, A154404, A154940.
%K nice,nonn
%O 1,9
%A _Zhi-Wei Sun_, Jan 30 2009
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