A071354 Floor(2^n/n) is odd.

12, 18, 25, 36, 42, 45, 48, 55, 80, 91, 95, 98, 99, 100, 108, 110, 112, 125, 130, 132, 135, 136, 140, 143, 152, 153, 155, 160, 161, 162, 175, 184, 187, 190, 192, 198, 208, 216, 224, 225, 228, 232, 235, 238, 240, 242, 245, 247, 248, 261, 266, 273, 275, 279, 285, 286, 289

Offset: 1

Author

R. K. Guy, Jun 12 2002

Keywords

• nonn
• easy

Comments

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.

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

Crossrefs

Cf. A000799, A071941.

Programs

Mathematica

Select[ Range[300], OddQ[ Floor[2^# / # ]] & ]

PARI

for(n=1,1000,if((-1)^(floor(2^n/n))==-1+isprime(n),print1(n,",")))

Python

from itertools import count, islice
from sympy import isprime
def A071354_gen(startvalue=1): # generator of terms >= startvalue
    yield from filter(lambda k:not isprime(k) and (1<A071354_list = list(islice(A071354_gen(),20)) # Chai Wah Wu, Apr 23 2025

Extensions

More terms from several correspondents, Jun 12 2002