A213982 - OEIS

%I #55 Mar 28 2018 23:15:30

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,2,1,1,3,1,2,

%T 1,1,1,1,1,1,1,2,1,1,2,2,1,1,1,2,1,2,2,3,1,1,1,1,1,2,1,1,3,2,1,1,1,3,

%U 1,2,1,1,1,3

%N Least k >= 1 such that prime(n) +- k = 2^m * q, m >= 0, where q >= 2 is prime.

%C What one can say about the average behavior of a(n) for large n? It is interesting in view of the Broughan-Qizhi inequality A192869(n) >> n*(log(n))^2 and their conjecture that A192869(n) = O(n*(log(n))^2). But in the case of A213982 we have, on average, log(n) possible odd positive and negative values of k with |k| < min (p_n-p_(n-1, p_(n+1)-p_n) which is approximately log(n).

%C Therefore, we conjecture that, on average, a(n) is approximately c*log(n) with c in (0,1). Calculations up to 10^6 (_Peter J. C. Moses_) show that, most likely, c < 0.298.

%H Robert Israel, <a href="/A213982/b213982.txt">Table of n, a(n) for n = 1..10000</a>

%H Kevin Broughan and Zhou Qizhi, <a href="http://www.math.waikato.ac.nz/~kab/papers/flatandthin4.pdf">Flat primes and thin primes</a>, Bulletin of the Australian Mathematical Society 82:2 (2010), pp. 282-292.

%e a(1)=1, since 2+1 = 3 = 2^0*3;

%e a(2)=1, since 3+1 = 2^1*2;

%e a(7)=1, since 17-1 = 16 = 2^3*2;

%e a(10)=1, since 29-1 = 28 = 2^2*7.

%p f:= proc(n) local p, q,k,t;

%p   p:= ithprime(n);

%p   for k from 1 do

%p     for t in [p+k,p-k] do

%p       q:= t/2^padic:-ordp(t,2);

%p       if q=1 or isprime(q) then return k fi

%p     od

%p   od

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Mar 27 2018

%t Table[NestWhile[#+1&, 1, Not[Apply[Or, Flatten[PrimeQ[Map[(Prime[n] + #)/(2^Range[0, Floor[Log[Prime[n]]/Log[2]]])&,{-#,#}]]]]]&], {n, 100}] (* _Peter J. C. Moses_, Jul 09 2012 *)

%Y Cf. A192869, A214018.

%K nonn

%O 1,20

%A _Vladimir Shevelev_, Jun 30 2012

%E Name edited by _Robert Israel_, Mar 28 2018