Almost prime

Demonstration, with Cuisenaire rods, of the 2-almost prime nature of the number 6

In number theory, a natural number is called almost prime if there exists an absolute constant K such that the number has at most K prime factors.^[1]^[2] An almost prime n is denoted by P_r if and only if the number of prime factors of n, counted according to multiplicity, is at most r.^[3] A natural number is called k-almost prime if it has exactly k prime factors, counted with multiplicity. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n:

Omega (n):=sum a_{i}qquad {mbox{if}}qquad n=prod p_{i}^{a_{i}}.

A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of k-almost primes is usually denoted by P_k. The smallest k-almost prime is 2^k. The first few k-almost primes are:

k	k-almost primes	OEIS sequence
1	2, 3, 5, 7, 11, 13, 17, 19, …	A000040
2	4, 6, 9, 10, 14, 15, 21, 22, …	A001358
3	8, 12, 18, 20, 27, 28, 30, …	A014612
4	16, 24, 36, 40, 54, 56, 60, …	A014613
5	32, 48, 72, 80, 108, 112, …	A014614
6	64, 96, 144, 160, 216, 224, …	A046306
7	128, 192, 288, 320, 432, 448, …	A046308
8	256, 384, 576, 640, 864, 896, …	A046310
9	512, 768, 1152, 1280, 1728, …	A046312
10	1024, 1536, 2304, 2560, …	A046314
11	2048, 3072, 4608, 5120, …	A069272
12	4096, 6144, 9216, 10240, …	A069273
13	8192, 12288, 18432, 20480, …	A069274
14	16384, 24576, 36864, 40960, …	A069275
15	32768, 49152, 73728, 81920, …	A069276
16	65536, 98304, 147456, …	A069277
17	131072, 196608, 294912, …	A069278
18	262144, 393216, 589824, …	A069279
19	524288, 786432, 1179648, …	A069280
20	1048576, 1572864, 2359296, …	A069281

The number π_k(n) of positive integers less than or equal to n with at most k prime divisors (not necessarily distinct) is asymptotic to:^[4]

pi _{k}(n)sim left({frac {n}{log n}}right){frac {(log log n)^{k-1}}{(k-1)!}},

a result of Landau. See also the Hardy–Ramanujan theorem.

References[edit]

^ Sándor, József; Dragoslav, Mitrinović S.; Crstici, Borislav (2006). Handbook of Number Theory I. Springer. p. 316. ISBN 978-1-4020-4215-7..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}

^ Rényi, Alfréd A. (1948). "On the representation of an even number as the sum of a single prime and single almost-prime number". Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya (in Russian). 12 (1): 57–78.

^ Heath-Brown, D. R. (May 1978). "Almost-primes in arithmetic progressions and short intervals". Mathematical Proceedings of the Cambridge Philosophical Society. 83 (3): 357–375. doi:10.1017/S0305004100054657.

^ Tenenbaum, Gerald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press. ISBN 0-521-41261-7.

External links[edit]

Weisstein, Eric W. "Almost prime". MathWorld.

[1] Sándor, József; Dragoslav, Mitrinović S.; Crstici, Borislav (2006). Handbook of Number Theory I. Springer. p. 316. ISBN 978-1-4020-4215-7..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}

[2] Rényi, Alfréd A. (1948). "On the representation of an even number as the sum of a single prime and single almost-prime number". Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya (in Russian). 12 (1): 57–78.

[3] Heath-Brown, D. R. (May 1978). "Almost-primes in arithmetic progressions and short intervals". Mathematical Proceedings of the Cambridge Philosophical Society. 83 (3): 357–375. doi:10.1017/S0305004100054657.

[4] Tenenbaum, Gerald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press. ISBN 0-521-41261-7.

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers Mathematics portal

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