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%I #8 Oct 21 2018 13:03:55
%S 12,18,25,36,42,45,48,55,80,91,95,98,99,100,108,110,112,125,130,132,
%T 135,136,140,143,152,153,155,160,161,162,175,184,187,190,192,198,208,
%U 216,224,225,228,232,235,238,240,242,245,247,248,261,266,273,275,279,285,286,289
%N Floor(2^n/n) is odd.
%C A student asked if the floor of 2^n / n was always even. He had a proof when n is prime. There is a shorter proof if you look at the binomial expansion of (1+1)^p.
%C There are infinitely many numbers in this sequence. (Because if n is even, then 2^n*12-n-2 is even, so 2^(2^n*12-n-2) is 4 (mod 6). Define x so that this is 6*x + 4, then dividing by 3 gives 2*x + (4/3), and the floor is an odd number.) - _Jinyuan Wang_, Oct 13 2018
%t Select[ Range[300], OddQ[ Floor[2^# / # ]] & ]
%o (PARI) for(n=1,1000,if((-1)^(floor(2^n/n))==-1+isprime(n),print1(n,",")))
%Y Cf. A000799, A071941.
%K nonn,easy
%O 1,1
%A _R. K. Guy_, Jun 12 2002
%E More terms from several correspondents, Jun 12, 2002
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