A round number is a number that is the product of a considerable number of comparatively small factors (Hardy 1999, p. 48). Round numbers are very rare. As Hardy (1999,
p. 48) notes, "Half the numbers are divisible by 2, one-third by 3, one-sixth
by both 2 and 3, and so on. Surely, then we may expect most numbers to have a large
number of factors. But the facts seem to show the opposite."
A positive integer
is sometimes said to be round (or "square root-smooth") if it has no prime
factors greater than .
The first few such numbers are 1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, ... (OEIS
A048098). Using this definition, an asymptotic
formula for the number of round integers less than or equal to a positive real number
is given by
(Hildebrand).
A different meaning of "round" is used when speaking of "rounding
a number."
is sometimes said to be round (or "square root-smooth") if it has no prime
factors greater than 
.
The first few such numbers are 1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, ... (OEIS
A048098). Using this definition, an asymptotic
formula for the number of round integers less than or equal to a positive real number
is given by
