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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