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Percentage of integers with lpf (*n*) = *p*.

lpf = 2 (50%)

lpf = 3 (16.666666666667%)

lpf = 5 (6.6666666666667%)

lpf = 7 (3.8095238095238%)

lpf = 11 (2.0779220779221%)

lpf = 13 (1.5984015984016%)

lpf = 17 (1.1282834812247%)

lpf = 19 (0.9501334578734%)

lpf = 23 (0.74358270616179%)

lpf = 29 (0.56409722536412%)

lpf = 31 (0.50950717129662%)

lpf = 37 (0.41311392267294%)

lpf = 41 (0.3627341760055%)

lpf = 43 (0.33742714047024%)

lpf ≥ 47 (14.17193989975%)

The

**least prime factor** of an integer

is the smallest

prime number that divides the number. For example, the least prime factor of

945 is

3. The least prime factor of all

even numbers is

2. A prime number is its own least prime factor (as well as its own

greatest prime factor).

By convention, 1 is given as its own least prime factor, but of course this has met with objections. By disallowing 1 as a prime number, we can then say that each prime number is its own least and greatest prime factor. However, in the OEIS, it is reasonable to believe that some users will look up the sequence of least prime factors as “1, 2, 3, 2, 5, 2, 7, 2, 3, 2 ” (give or take a few terms), and that should deliver a result.

Smallest prime dividing

. (See

A020639 Lpf (*n*):

Least prime dividing *n*, with

.)

{2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, ...}

## Density of integers with smallest prime factor prime (*n*)

The density of positive integers with smallest prime factor prime (*n*) is

{ , , , , , , , , , ...} = |

where

{1, , , , , , , , , ...} = |

{1, 1 − , 1 − − , 1 − − − , 1 − − − − , 1 − − − − − , ...} |

is the density of prime (*n*)-rough numbers.

The density of positive integers with smallest prime factor prime (*n*) is equal to the density of prime (*n*)-rough numbers minus the density of prime (*n* + 1)-rough numbers.

A038110 Numerator of

. Numerator of density of integers with smallest prime factor

prime (*n*). Numerator of density of

prime (*n*)-rough numbers.

{1, 1, 1, 4, 8, 16, 192, 3072, 55296, 110592, 442368, 13271040, 477757440, 19110297600, 802632499200, 1605264998400, 6421059993600, 12842119987200, 770527199232000, 50854795149312000, 3559835660451840000, ...}

A038111 Denominator of

.

*a* (*n*) = A060753 (*n*) ⋅ prime (*n*). Denominator of density of integers with smallest prime factor

prime (*n*).

{2, 6, 15, 105, 385, 1001, 17017, 323323, 7436429, 19605131, 86822723, 3212440751, 131710070791, 5663533044013, 266186053068611, 613385252723321, 2783825377744303, 5855632691117327, 392327390304860909, ...}

A060753 Denominator of

.

*a* (*n*) = A038111 (*n*) / prime (*n*). Denominator of density of

prime (*n*)-rough numbers.

{1, 2, 3, 15, 35, 77, 1001, 17017, 323323, 676039, 2800733, 86822723, 3212440751, 131710070791, 5663533044013, 11573306655157, 47183480978717, 95993978542907, 5855632691117327, 392327390304860909, ...}

## See also