P-smooth numbers

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A

p

-smooth number, or

p

-friable number, is an integer whose largest prime factor is

p

, or an integer without any prime factor larger than

p

(defined thusly, it allows the inclusion of 1, the empty product of primes).

The 2-smooth numbers (  p1-smooth numbers) among the integers (excluding zero) are the units −1 and +1 and ± the powers of 2 with positive exponent, thus ± the powers of 2 with nonnegative exponent.

Among the positive integers (greater than 1):

• 2-smooth numbers (  p1-smooth numbers):

  ∏ 1 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   1

  ;

• 3-smooth numbers (  p2-smooth numbers):

  ∏ 2 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   2

  ;

• 5-smooth numbers (  p3-smooth numbers):

  ∏ 3 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   3

  ;

• 7-smooth numbers (  p4-smooth numbers):

  ∏ 4 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   4

  ;

• 11-smooth numbers (  p5-smooth numbers):

  ∏ 5 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   5

  ;

• 13-smooth numbers (  p6-smooth numbers):

  ∏ 6 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   6

  ;

• 17-smooth numbers (  p7-smooth numbers):

  ∏ 7 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   7

  ;

• 19-smooth numbers (  p8-smooth numbers):

  ∏ 8 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   8

  ;

• 23-smooth numbers (  p9-smooth numbers):

  ∏ 9 i  = 1 pi ei , ei   ≥   1, 1   ≤   i   ≤   9

  ;

• ...

p	p -smooth numbers*	A-number
2	{1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, ...}	A000079
3	{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972, ...}	A003586
5	{1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, ...}	A051037
7	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, ...}	A002473
11	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, ...}	A051038
13	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, ...}	A080197
17	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 60, ...}	A080681
19	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, ...}	A080682
23	{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, ...}	A080683
29	{1, ...}	A??????
31	{1, ...}	A??????

* The OEIS considers 1 as a

p

-smooth number for any

p

, although it doesn’t have a smallest prime factor (it is the empty product of primes).

The density of prime (n)-smooth numbers is zero.

Sequences

A063539

2√  n  −  1

-smooth numbers: largest prime factor of

n

(A006530)

< 2√  n

.

{1, 8, 12, 16, 18, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 132, 135, 140, 144, 147, ...}

See also

• a b c conjecture
• Largest prime factor of n
• p-rough numbers

External links

• Carl Pomerance, The Role of Smooth Numbers in Number Theoretic Algorithms.