The least nondivisor of a positive integer 
is the smallest positive
integer that does not divide 
. Writing this number as 
gives
 |
(1)
|
For 
,
2, ..., its values begin 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, ... (OEIS A007978).
In particular, 
for odd 
and 
for even 
.
The possible values are exactly the prime powers 
with 
. If 
had two distinct prime factors,
each of its maximal prime power factors would be smaller
than 
and would divide 
. Their product would then divide 
, contrary to the definition. Conversely,
the first occurrence of a prime power 
is at
 |
(2)
|
since this least common multiple is divisible by every smaller positive integer but has exponent
at 
(OEIS A007978).
Put 
for 
and 
.
The exact distribution follows from
 |
(3)
|
so the number of positive integers 
with 
is 
, where 
denotes the floor function.
Consequently, the natural density of the set with
is
 |
(4)
|
This is positive precisely when 
is a prime power. Farhi (2009,
p. 556) proved the average-order formula
 |
(5)
|
More generally, the 
th
least nondivisor is the 
th smallest positive integer
that does not divide 
. The second least nondivisors begin 3, 4, 4, 5, 3, 5, 3, 5,
4, 4, ... (OEIS A396771). Their possible values
are the integers 
that are either a prime power
or twice a prime power (Radcliffe 2026). Indeed, if
but 
, then every multiple
of 
through 
fails to divide 
. There can be at most two such multiples
when 
is the second least nondivisor, forcing 
.
