A213982 Least k >= 1 such that prime(n) +- k = 2^m * q, m >= 0, where q >= 2 is prime.

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, 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, 1, 2, 1, 1, 1, 3

Offset: 1

Author

Vladimir Shevelev, Jun 30 2012

Keywords

• nonn

Comments

What can one 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(prime(n)-prime(n-1), prime(n+1)-prime(n)) which is approximately log(n).

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.

Examples

			a(1) = 1, since 2+1 = 3 = 2^0*3;
a(2) = 1, since 3+1 = 2^1*2;
a(7) = 1, since 17-1 = 16 = 2^3*2;
a(10) = 1, since 29-1 = 28 = 2^2*7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Kevin Broughan and Zhou Qizhi, Flat primes and thin primes, Bulletin of the Australian Mathematical Society 82:2 (2010), pp. 282-292.

Crossrefs

Cf. A192869, A214018.

Programs

Maple

f:= proc(n) local p, q,k,t;
  p:= ithprime(n);
  for k from 1 do
    for t in [p+k,p-k] do
      q:= t/2^padic:-ordp(t,2);
      if q=1 or isprime(q) then return k fi
    od
  od
end proc:
map(f, [$1..100]); # Robert Israel, Mar 27 2018

Mathematica

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 *)

Extensions

Name edited by Robert Israel, Mar 28 2018