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Prime signatures
A positive integer with prime factorization
{α1, α2, …, αk } 
{…} 
1 
{ } 
{1} 
p k 
{k} 
Goldston, Graham, Pintz, & Yıldırım call these exponent patterns,^{[1]} and King refers to them as prime bags (PBs).^{[2]}
The prime signature ([unordered] prime signature) ofn 
n 
12 = 2 2 × 3 
18 = 2 × 3 2 
{2, 1} 
12 
(2, 1) 
18 
(1, 2) 
(…) 
Contents
Prime signature as partition of Omega(n)
Numbers and their prime signatures  


Factorization  Prime signature  
1  { }  
2  {1}  
3  {1}  
4  {2}  
5  {1}  
6  {1,1}  
7  {1}  
8  {3}  
9  {2}  
10  {1,1}  
11  {1}  
12  {2,1}  
13  {1}  
14  {1,1}  
15  {1,1}  
16  {4}  
17  {1}  
18  {2,1}  
19  {1}  
20  {2,1}  
21  {1,1}  
22  {1,1}  
23  {1}  
24  {3,1}  
25  {2}  
26  {1,1}  
27  {3}  
28  {2,1}  
29  {1}  
30  {1,1,1} 
n 
Ω (n) 
Ω (n) 
n 
Number of divisors and prime signature
Since the number of divisors depends only on the exponents in the prime factorization of a number, all numbers of a given prime signature have the same number of divisors.
Prime signatures and their numbers
See Orderings of prime signatures.
Special numbers and their prime signatures
A number with prime signature
is
 A unit if
;S = { }  A prime if
;S = {1}  A prime power if
;S = {k}, k ≥ 2  A square if
is even;gcd (S)  A kth power if
is a multiple ofgcd (S)
;k, k ≥ 2  A square of a squarefree number if
;min (S) = max (S) = 2  A
th power of a squarefree number ifk
;min (S) = max (S) = k, k ≥ 2  A squarefree number if
;max (S) = 1  A squareful number (2powerful number) if
;max (S) ≥ 2  A squarefull number (2powerfull number) if
;min (S) ≥ 2  A cubeful number (3powerful number) if
;max (S) ≥ 3  A cubefull number (3powerfull number) if
;min (S) ≥ 3  A biquadrateful number (4powerful number) if
;max (S) ≥ 4  A biquadratefull number (4powerfull number) if
;min (S) ≥ 4  A kpowerful number if
;max (S) ≥ k, k ≥ 2  A kpowerfull number if
;min (S) ≥ k, k ≥ 2  A kalmost prime if
;αi = k  A squarefree kalmost prime if
andαi = k
;max (S) = 1  An Achilles number if
andmin (S) ≥ 2
.gcd (S) = 1
Sequences
A118914 Concatenation of the prime signatures (in increasing order of exponents of prime power components) of the positive integers.

{1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 2, 1, 2, 1, ...}
A025487 Least integer of each prime signature; also products of primorial numbers A002110.

{1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, ...}
A036035 Least integer of each prime signature, in graded (colexicographic order or reflected colexicographic order) of exponents.

{1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, ...}
n 

{1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, ...}
1 

{2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 18, 16, 17, 169, 21, 343, 20, 81, 24, 19, 289, 22, 1331, 28, 625, 40, 30, 23, 361, 26, 2197, 44, 2401, 54, 42, 32, 29, 529, 33, 4913, 45, ...}
See also
 Prime factorization
 Orderings of prime signatures
 Ordered prime signatures (where the order of exponents matters, corresponding to compositions of
)Ω (n)  Orderings of ordered prime signatures
Notes
 ↑ D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yıldırım, Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers, International Mathematics Research Notices 7 (2011), pp. 14391450.
 ↑ Ross D. King, Numbers as data structures: the prime successor function as primitive, 2011.
External links
 Will Nicholes, List of the first 400 prime signatures.
 Will Nicholes, Iterative mapping of prime signatures.