cp's OEIS Frontend

A155860 Number of ways to write 2n-1 as p + 2^x + 3*2^y with p an odd prime and x,y positive integers.

%I A155860 #22 Sep 10 2018 14:09:01
%S A155860 0,0,0,0,0,1,2,2,3,4,5,3,5,7,4,7,9,5,6,9,5,7,11,6,6,12,5,9,13,8,10,12,
%T A155860 4,11,15,6,10,15,5,9,16,9,9,17,8,8,17,8,10,16,8,11,13,10,10,20,7,12,
%U A155860 23,10,10,21,9,11,18,11,8,18,9,11,20,9,13,17,9,12,19,9,13,22,6,13,21,10,10,21
%N A155860 Number of ways to write 2n-1 as p + 2^x + 3*2^y with p an odd prime and x,y positive integers.
%C A155860 On Jan 21 2009, _Zhi-Wei Sun_ conjectured that a(n)>0 for n=6,7,...; in other words, any odd integer m>10 can be written as the sum of an odd prime, a positive power of 2 and three times a positive power of 2. Sun verified this for odd integers m<10^7. On Sun's request, Qing-Hu Hou and _Charles R Greathouse IV_ continued the verification for odd integers below 2*10^8 and 10^10 respectively and found no counterexamples.
%C A155860 As 3*2^y = 2^y + 2^{y+1}, Sun's conjecture implies that each odd integer m>8 can be written as the sum of an odd prime and three positive powers of two. Note that Paul Erdős asked whether there is a positive integer r such that every odd integer m>3 can be written as the sum of a prime and at most r powers of 2.
%C A155860 _Zhi-Wei Sun_ also raised the following problem: For k=3,5,...,61 determine whether any odd integer m>2k+3 can be written in the form p + 2^x + k*2^y with p an odd prime and x,y positive integers. Sun observed that 353 is not of the form p + 2^x + 51*2^y and Qing-Hu Hou continued the search for m<2.5*10^7 and found that 22537515 is not of the form p + 2^x + 47*2^y. For k=3,5,...,45,49,53,55,...,61, Sun has checked odd integers below 10^8 and found no odd integer m>2k-3 not of the form p + 2^x + k*2^y.
%D A155860 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
%D A155860 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 A155860 Zhi-Wei Sun, <a href="/A155860/b155860.txt">Table of n, a(n) for n = 1..50000</a>
%H A155860 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 A155860 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 A155860 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 A155860 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 A155860 a(n) = |{<p,x,y>: p+2^x+3*2^y = 2n-1 with p an odd prime and x,y positive integers}|.
%e A155860 For n=10 the a(10)=4 solutions are 19 = 3 + 2^2 + 3*2^2 = 5 + 2 + 3*2^2 = 5 + 2^3 + 3*2 = 11 + 2 + 3*2.
%t A155860 PQ[x_]:=x>2&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-3*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/3]},{y,1,Log[2,Max[2,2n-1-3*2^x]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]
%Y A155860 Cf. A154257, A154285, A155114, A154536, A154404, A154940.
%K A155860 nice,nonn
%O A155860 1,7
%A A155860 _Zhi-Wei Sun_, Jan 29 2009