Search: id:A157237
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%I A157237
%S A157237 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,2,1,0,2,2,0,1,1,1,2,2,2,4,1,2,5,2,
%T A157237 1,3,1,1,2,1,3,3,1,3,5,2,2,5,4,0,5,4,2,4,3,3,4,3,3
%N A157237 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y 
               with p a prime congruent to 1 mod 6 and x,y positive integers.
%C A157237 On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if 
               n<16 or n=18, 21, 24, 51, 84, 1011, 59586; in other words, except 
               for 35, 41, 47, 101, 167, 2021, 119171, any odd integer greater than 
               30 can be written as the sum of a prime congruent to 1 mod 6, a positive 
               power of 2 and eleven 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 A157237 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 
               36(1971), 103-107.
%D A157237 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.
%D A157237 Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. http:/
               /arxiv.org/abs/0901.3075
%H A157237 Zhi-Wei Sun, <a href="b157237.txt">Table of n, a(n) for n=1..200000</
               a>
%H A157237 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 A157237 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 A157237 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>
%F A157237 a(n)=|{<p,x,y>: p+2^x+11*2^y=2n-1 with p a prime congruent to 1 mod 6 
               and x,y positive integers}|
%e A157237 For n=19 the a(19)=2 solutions are 2*19-1=7+2^3+2*11=13+2+2*11.
%t A157237 PQ[x_]:=x>1&&Mod[x,6]==1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-11*2^x-2^y],
               1,0], {x,1,Log[2,(2n-1)/11]},{y,1,Log[2,Max[2,2n-1-11*2^x]]}] Do[Print[n,
               " ",RN[n]],{n,1,200000}]
%Y A157237 A000040, A000079, A157218, A157225, A155860, A155904, A156695, A154257, 
               A154285, A155114, A154536
%Y A157237 Sequence in context: A134663 A000925 A003985 this_sequence A065676 A146973 
               A003263
%Y A157237 Adjacent sequences: A157234 A157235 A157236 this_sequence A157238 A157239 
               A157240
%K A157237 nonn
%O A157237 1,19
%A A157237 Zhi-Wei Sun (zwsun(AT)nju.edu.cn), Feb 25 2009

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