Ordered prime signatures

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A positive integer with prime factorization

${p_{1}}^{\alpha _{1}}{p_{2}}^{\alpha _{2}}\cdots {p_{k}}^{\alpha _{k}},\quad p_{1}<p_{2}<\cdots <p_{k},$

is said to have an ordered prime signature expressed as the [[Tuples| 

k

-tuple]] 

(α1, α2, …, αk )

. For example, 

12

has ordered prime signature 

(2, 1)

which does not equal 

(1, 2)

, the ordered prime signature of 

18

. Ordered prime signatures correspond to compositions of 

Ω (n)

while prime signatures correspond to partitions of 

Ω (n)

.

Orderings of ordered prime signatures

Cf. Orderings of ordered prime signatures.

Sequences

A055932 Least integer of each ordered prime signatures.

 

{1, 2, 4, 6, 8, 12, 16, 18, 24, 30, 32, 36, 48, 54, 60, 64, 72, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192, 210, 216, 240, 256, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 512, 540, 576, ...}

A096903 Least integer of each ordered prime signatures (A055932) arranged by prime signature (each row starting with least integer of each prime signature, A025487).

 

{1, 2, 4, 6, 8, 12, 18, 16, 24, 54, 30, 32, 36, 48, 162, 60, 90, 150, 64, 72, 108, 96, 486, 120, 270, 750, 128, 144, 324, 180, 300, 450, 192, 1458, 210, 216, 240, 810, 3750, 256, 288, 972, 360, 540, 600, ...}

A071364 Smallest number with same sequence of exponents in canonical prime factorization as 

n

.

 

{1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 18, 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, 18, 6, 12, 2, 54, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, ...}

See also

• Prime factorization
• Orderings of ordered prime signatures
• Prime signatures ([unordered] prime signatures, i.e. multiset of exponents)
• Orderings of prime signatures