Triprimes

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Note: The term " 

k

-almost prime," e.g. " 

3

-almost prime," is deprecated. Please use "product of exactly 

k

primes" instead.

Conway et al. (2008) propose calling " 

k

-almost primes" thusly (i.e. using Latin prefixes): primes, biprimes, triprimes, quadruprimes, quinqueprimes, sextiprimes, ...

Index to OEIS: Triprimes

Triprimes (  

p q r

numbers) are composite numbers 

n = p q r

with exactly three [not necessarily distinct] prime factors 

p   ≤   q   ≤   r

. The set of triprimes is denoted 

P3

.

A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

 

{8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, ...}

A101605 Characteristic function of triprimes: 

a (n) = 1

if 

n

is a product of exactly 

3

primes, otherwise 

0

.

 

{0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, ...}

A114403 Triprimes gaps: differences between triprimes. First differences of A014612.

 

{4, 6, 2, 7, 1, 2, 12, 2, 1, 5, 2, 11, 3, 2, 2, 5, 1, 2, 14, 6, 1, 3, 3, 5, 4, 2, 1, 7, 1, 5, 8, 9, 1, 5, 1, 10, 1, 5, 1, 1, 2, 1, 7, 4, 2, 2, 5, 12, 5, 10, 8, 1, 5, 2, 4, 2, 1, 1, 9, 3, 3, 5, 2, 5, 2, 4, 3, 2, 1, 1, 4, 2, 18, 6, 2, 4, ...}

A?????? Prime factor 

p   ≤

other prime factors 

q

and 

r

of 

n

-th triprime.

 

{2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, ...}

A162361 Central prime factor of triprimes.

 

{2, 2, 3, 2, 3, 2, 3, 3, 2, 3, 5, 2, 3, 3, 2, 5, 5, 2, 3, 2, 7, 3, 3, 5, 5, 3, 2, 3, 2, 5, 5, 3, 7, 2, 3, 7, 2, 5, 5, 3, 2, 3, 5, 7, 3, 2, 5, 5, 3, 2, 3, 5, 7, 2, 7, 11, 2, 7, 3, 5, 3, 3, 7, 2, 7, 5, 3, 3, 2, 5, 11, 5, 2, 5, 2, 3, 7, 5, 2, 3, ...}

A?????? Prime factor 

r   ≥

other prime factors 

p

and 

q

of 

n

-th triprime.

 

{2, 3, 3, 5, 3, 7, 5, 7, 11, 5, 5, 13, 7, 11, 17, 7, 5, 19, 13, 23, 7, 11, 17, 7, 11, 19, ...}

A102304 Sum of factors of numbers having exactly three prime factors.

 

{6, 7, 8, 9, 9, 11, 10, 12, 15, 11, 12, 17, 13, 16, 21, 14, 13, 23, 18, 27, 16, 17, 22, 15, 18, 24, 33, 19, 35, 15, 20, 28, 17, 41, 23, 20, 45, 19, 24, 25, 47, 34, 17, 22, 36, 51, 26, 21, 29, 57, 42, 30, 21, 63, 26, 24, ...}

Arithmetic progressions and triprimes

A075818 Even numbers with exactly 

3

prime factors (counted with multiplicity).

 

{8, 12, 18, 20, 28, 30, 42, 44, 50, 52, 66, 68, 70, 76, 78, 92, 98, 102, 110, 114, 116, 124, 130, 138, 148, 154, 164, 170, 172, 174, 182, 186, 188, 190, 212, 222, 230, 236, 238, 242, 244, 246, 258, 266, ...}

A046316 Odd numbers divisible by exactly 

3

primes (counted with multiplicity).

 

{27, 45, 63, 75, 99, 105, 117, 125, 147, 153, 165, 171, 175, 195, 207, 231, 245, 255, 261, 273, 275, 279, 285, 325, 333, 343, 345, 357, 363, 369, 385, 387, 399, 423, 425, 429, 435, 455, 465, 475, ...}

Nonsquarefree triprimes

A030078 Cubic triprimes: cubes of primes.

 

{8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 912673, 1030301, 1092727, ...}

Cubic triprimes have the following properties:

•  

  d ( p 3) = 4

  : they have exactly four divisors;
•  

  φ ( p 3) = ( p  −  1)  p 2

  .

A054753 Numbers which are the product of a prime and the square of a different prime.

 

{12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, ...}

Numbers which are the product of a prime and the square of a different prime have the following properties:

•  

  d ( p 2 q) = 6, p   ≠   q

  : they have exactly six divisors;
•  

  φ ( p 2 q) = ( p  −  1)  p  (q  −  1), p   ≠   q

  .

Squarefree triprimes

Main article page: Sphenic number

A007304 Sphenic numbers: products of 

3

distinct primes.

 

{30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, ...}

A075819 Even sphenic numbers: even squarefree numbers with exactly 

3

prime factors.

 

{30, 42, 66, 70, 78, 102, 110, 114, 130, 138, 154, 170, 174, 182, 186, 190, 222, 230, 238, 246, 258, 266, 282, 286, 290, 310, 318, 322, 354, 366, 370, 374, 402, 406, 410, 418, 426, 430, 434, 438, 442, 470, ...}

A046389 Odd sphenic numbers: products of 

3

distinct odd primes.

 

{105, 165, 195, 231, 255, 273, 285, 345, 357, 385, 399, 429, 435, 455, 465, 483, 555, 561, 595, 609, 615, 627, 645, 651, 663, 665, 705, 715, 741, 759, 777, 795, 805, 861, 885, 897, 903, 915, 935, 957, 969, ...}

Squarefree triprimes have the following properties:

•  

  d ( p q r) = 8, p < q < r

  : they have exactly eight divisors;
•  

  φ ( p q r) = ( p  −  1) (q  −  1) (r  −  1), p < q < r

  .

Triprime counting function

A072114 Triprime counting function: number of triprimes (A014612) 

≤   n, n   ≥   1.

 

{0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, ...}

A109251 Number of triprimes 

< 10 n, n   ≥   0

.

 

{0, 1, 22, 247, 2569, 25556, 250853, 2444359, 23727305, 229924367, 2227121996, 21578747909, 209214982913, 2030133769624, 19717814526785, 191693417109381, 1865380637252270, ...}

n	π3(n)	n log n ( log log n) 2 2
10 1	1	1.51
10 2	22	25.32
10 3	247	270.36
10 4	2569	2676.26
10 5	25556	25929.76
10 6	250853	249530.52
10 7	2444359	2397330.61
10 8	23727305	23040219.82

The triprime counting function is asymptotic to (which is unfortunately a very poor approximation)

π3(n) ∼ π2(n) log log n 2 ∼ π (n) ( log log n) 2 2 ∼ n log n ( log log n) 2 2 ,

where 

π2(n)

is the biprime counting function and 

π (n)

is the prime counting function.

Totient of triprimes

A?????? Totient of the 

n

-th triprime.

 

{6, 8, 18, 12, 8, 12, 20, 24, 20, 24, 36, 20, 32, ...}

Triprimality testing

(...)

See also

• Biprimes

External links

• Conway, J. H.; Dietrich, H.; O'Brien, E. A. (2008). “Counting Groups: Gnus, Moas, and Other Exotica”. Math. Intell. 30: pp. 6–18.