The least prime factor 
(also denoted 
) of an integer 
is the least integer greater
than 1 that divides 
,
i.e., the number 
in the prime factorization
 |
with 
for 
.
The least prime factor is implemented in the Wolfram
Language as FactorInteger[n][[1,1]].
For 
,
3, ..., the first few are 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, ... (OEIS A020639).
If 
is a composite number then ![[lpf(n)]^2<=n](/images/equations/LeastPrimeFactor/Inline10.svg)
(Séroul 2000, p. 7), with equality
iff 
for a prime 
.
A plot of the least prime factor function resembles a jagged terrain of mountains, which leads to the appellation of "twin peaks"
to a pair of integers 
with 
such that
1. 
,
2. For every integer 
, 
implies 
.
For a squareful integer 
, the least multiple prime
factor is the smallest prime 
with 
dividing 
. This may differ from 
, as it does for 
. The first values are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2,
2, 2, ... (OEIS A046027).
Erdős et al. (1993) consider the least prime factor of the binomial coefficients, and define what they term good
binomial coefficients and exceptional
binomial coefficients. For integers 
and 
with 
, they conjecture that
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