Least Prime Factor

For an Integer $n\geq 2$, let $\mathop{\rm lpf}\nolimits (x)$ denote the Least Prime Factor of $n$, i.e., the number $p_1$ in the factorization

$$n={p_1}^{a_1}\cdots{p_k}^{a_k},$$

with $p_i<p_j$ for $i<j$. For $n=2$, 3, ..., the first few are 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, ... (Sloane's A020639). The above plot of the least prime factor function can be seen to resemble a jagged terrain of mountains, which leads to the appellation of ``Twin Peaks'' to a Pair of Integers $(x, y)$ such that

1. $x < y$,

2. $\mathop{\rm lpf}\nolimits (x) = \mathop{\rm lpf}\nolimits (y)$,

3. For all $z$, $x < z < y$ Implies $\mathop{\rm lpf}\nolimits (z) < \mathop{\rm lpf}\nolimits (x)$.

The least multiple prime factors for Squareful integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, ... (Sloane's A046027).

See also Alladi-Grinstead Constant, Distinct Prime Factors, Erdös-Selfridge Function, Factor, Greatest Prime Factor, Least Common Multiple, Mangoldt Function, Prime Factors, Twin Peaks

References

Sloane, N. J. A. A020639 and A046027 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.