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Prime signatures

(Redirected from Prime signature)

${\displaystyle p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}}$
is said to have a prime signature (an [unordered] prime signature) of
 {α1, α2, …, αk }
where
 {…}
is a multiset (or bag). For example, the prime signature of
 1
is the empty multiset
 { }
, primes have a prime signature
 {1}
, and prime powers
 p k
have prime signature
 {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) of
 n
should not be confused with the ordered prime signature of
 n
: for example
 12 = 2 2  ×  3
and
 18 = 2  ×  3 2
both have prime signature
 {2, 1}
while
 12
has ordered prime signature
 (2, 1)
and
 18
has ordered prime signature
 (1, 2)
, where
 (…)
is a tuple.

Prime signature as partition of Omega(n)

Numbers and their prime signatures
 n
Factorization Prime signature
1 { }
2 ${\displaystyle 2}$ {1}
3 ${\displaystyle 3}$ {1}
4 ${\displaystyle 2^{2}}$ {2}
5 ${\displaystyle 5}$ {1}
6 ${\displaystyle 2\cdot 3}$ {1,1}
7 ${\displaystyle 7}$ {1}
8 ${\displaystyle 2^{3}}$ {3}
9 ${\displaystyle 3^{2}}$ {2}
10 ${\displaystyle 2\cdot 5}$ {1,1}
11 ${\displaystyle 11}$ {1}
12 ${\displaystyle 2^{2}\cdot 3}$ {2,1}
13 ${\displaystyle 13}$ {1}
14 ${\displaystyle 2\cdot 7}$ {1,1}
15 ${\displaystyle 3\cdot 5}$ {1,1}
16 ${\displaystyle 2^{4}}$ {4}
17 ${\displaystyle 17}$ {1}
18 ${\displaystyle 2\cdot 3^{2}}$ {2,1}
19 ${\displaystyle 19}$ {1}
20 ${\displaystyle 2^{2}\cdot 5}$ {2,1}
21 ${\displaystyle 3\cdot 7}$ {1,1}
22 ${\displaystyle 2\cdot 11}$ {1,1}
23 ${\displaystyle 23}$ {1}
24 ${\displaystyle 2^{3}\cdot 3}$ {3,1}
25 ${\displaystyle 5^{2}}$ {2}
26 ${\displaystyle 2\cdot 13}$ {1,1}
27 ${\displaystyle 3^{3}}$ {3}
28 ${\displaystyle 2^{2}\cdot 7}$ {2,1}
29 ${\displaystyle 29}$ {1}
30 ${\displaystyle 2\cdot 3\cdot 5}$ {1,1,1}
The prime signature of
 n
is a partition of
 Ω (n)
${\displaystyle \Omega (n)=\sum _{i=1}^{\omega (n)}\alpha _{i},}$
where
 Ω (n)
is the number of prime factors (with repetition) of
 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.

Special numbers and their prime signatures

A number with prime signature

${\displaystyle S=\{\alpha _{1},\alpha _{2},...,\alpha _{\omega (n)-1},\alpha _{\omega (n)}\}}$

is

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, ...}
A046523 Smallest number with same prime signature as
 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, ...}
A095904 Triangular array of natural numbers (greater than
 1
) arranged by prime signature.
 {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, ...}