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Omni-MATH / We have a calculator with two buttons that displays an integer x. Pressing the first button replaces x by ⌊(x)/(2)⌋, and pressing the second button replaces x by 4x+1. Initially, t…

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problem

We have a calculator with two buttons that displays an integer xx. Pressing the first button replaces xx by ⌊x2⌋\left\lfloor\frac{x}{2}\right\rfloor, and pressing the second button replaces xx by 4x+14 x+1. Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, ⌊y⌋\lfloor y\rfloor denotes the greatest integer less than or equal to the real number yy.)
Plain-text mathematical notation (without MathML)
We have a calculator with two buttons that displays an integer x. Pressing the first button replaces x by ⌊(x)/(2)⌋, and pressing the second button replaces x by 4x+1. Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, ⌊y⌋ denotes the greatest integer less than or equal to the real number y.)
Original LaTeX notation
We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\left\lfloor\frac{x}{2}\right\rfloor$, and pressing the second button replaces $x$ by $4 x+1$. Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y\rfloor$ denotes the greatest integer less than or equal to the real number $y$.)

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Source and history

task_id
c16ad551-a878-5d52-b24b-608013f9fda5
task_key
test--c16ad551-a878-5d52-b24b-608013f9fda5
task_revision_id
1
config / split
/ test

Official source

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