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A092111 a(n) = n+1 minus the greatest number of 1's in the binary representations of primes between 2^n and 2^(n+1).

%I A092111 #13 Mar 05 2020 22:52:04
%S A092111 0,0,1,0,1,0,1,1,1,1,1,0,1,2,1,0,1,0,1,1,1,2,1,1,1,2,1,1,1,0,1,1,1,1,
%T A092111 1,2,1,2,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,0,1,2,1,1,1,1,1,1,
%U A092111 1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,0,1,2,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1
%N A092111 a(n) = n+1 minus the greatest number of 1's in the binary representations of primes between 2^n and 2^(n+1).
%C A092111 0's occur only at Mersenne prime exponents (A000043) - 1, twos are in A092112, threes do not appear < 504.
%C A092111 a(n) <= 2 for n <= 2000. - _Robert Israel_, Mar 05 2020
%H A092111 Robert Israel, <a href="/A092111/b092111.txt">Table of n, a(n) for n = 1..2000</a>
%F A092111 a(n) = n+1 - A091937(n).
%p A092111 f:= proc(n) local t,j,k;
%p A092111   t:= 2^(n+1)-1;
%p A092111   if isprime(t) then return 0 fi;
%p A092111   for j from 1 to n-1 do if isprime(t-2^j) then return 1 fi od;
%p A092111   for j from 1 to n-2 do for k from j+1 to n-1 do
%p A092111     if isprime(t-2^j-2^k) then return 2 fi od od;
%p A092111   FAIL
%p A092111 end proc:
%p A092111 map(f, [$1..200]); # _Robert Israel_, Mar 05 2020
%t A092111 Compute the second line of the Mathematica code for A091938, then (Table[n + 1, {n, 105}]) - (Count[ IntegerDigits[ #, 2], 1] & /@ Table[ f[n], {n, 105}])
%Y A092111 Cf. A091938, A092112.
%K A092111 nonn
%O A092111 1,14
%A A092111 _Robert G. Wilson v_, Feb 20 2004