A225889 - OEIS

%I #9 May 25 2013 22:30:34

%S 3,5,7,5,7,11,13,11,11,17,19,17,17,23,17,23,23,31,23,41,23,41,31,47,

%T 29,47,37,59,41,59,37,59,43,67,37,67,43,67,43,73,61,83,53,83,47,101,

%U 61,97,53,97,59,97,59,103,61,109,67,127,67,131

%N Least prime p_m such that n = p_m-p_{m-1}+...+(-1)^(m-k)*p_k for some 0<k<m, where p_j denotes the j-th prime.

%C By a conjecture of the author, a(n) <= 2*n+2.2*sqrt(n), and moreover a(n) <= n+4.6*sqrt(n) if n is odd. Clearly a(n)>n. We guess that a(2n)/(2n) --> 2 as n tends to the infinity.

%C Note that this sequence is different from A222579 which involves a stronger conjecture of the author.

%C Zhi-Wei Sun also conjectured that any positive even integer m can be written in the form p_n-p_{n-1}+...+(-1)^{n-k}*p_k with k < n and 2m-3.6*sqrt(m+1) < p_n < 2m+2.2*sqrt(m).

%H Zhi-Wei Sun, <a href="/A225889/b225889.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.

%e a(7) = 13 since 7 = 13-11+7-5+3.

%e a(20) = 41 since 20 = 41-37+31-29+23-19+17-13+11-7+5-3.

%t s[0_]:=0

%t s[n_]:=s[n]=Prime[n]-s[n-1]

%t Do[Do[If[s[j]-(-1)^(j-i)*s[i]==m,Print[m," ",Prime[j]];Goto[aa]],{j,PrimePi[m]+1,PrimePi[2m+2.2Sqrt[m]]},{i,0,j-2}];

%t Print[m," ",counterexample];Label[aa];Continue,{m,1,100}]

%Y Cf. A000040, A222579.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, May 19 2013