Divisorial

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The divisorial (divisor product) of

n

is the product of divisors of

n

, i.e. the product of all positive integers up to

n

which divide

n

. By analogy with the term “phi-torial” for the coprimorial of

n

(a product involving

φ (n)

, the number of positive integers up to and coprime to

n

), the divisorial of

n

might be called the “tau-torial” since the product involves

τ (n)

, the number of divisors of

n

.

Formulae

The divisorial of

n

is

τ ! (n)  =  Π τ (n)  :=    n ∏   i   = 1 i ∣n    i  =    n ∏   i   = 1    i [i ∣n]  =    n ∏   i   = 1    i [n mod i = 0],

where

τ (n)

is the number of divisors of

n

and [·] is the Iverson bracket.

The following identity holds

Π τ (n)  =  n  τ (n) 2  =  2√  n τ (n) ,

where

τ (n)

is the number of divisors of

n

. Note that

n τ (n)

is always a square since

τ (n)

is odd iff

n

is a square.
From the canonical prime factorization of

n

n  =    ω (n) ∏   i   = 1    pi αi,

the canonical prime factorization of

τ ! (n)

is

τ ! (n)  =    ω (n) ∏   i   = 1    pi βi,

with

βi  =  αi 2 τ (n)  =  αi 2   ω (n) ∏   j   = 1    (αj + 1).

Properties

All terms of this sequence occur only once.

Divisorial primes

The set of divisorial primes is the union of almost divisorial primes and quasi divisorial primes.

Almost divisorial primes

The almost divisorial primes are primes

p

such that

p  =      d ∣n ∏   d ∣n    d − 1  =      n ∏   i   = 1    d  [d ∣n] − 1

for some

n

and [·] is the Iverson bracket.

Quasi divisorial primes

The quasi divisorial primes are primes

p

such that

p  =      d ∣n ∏   d ∣n    d + 1  =      n ∏   i   = 1    d  [d ∣n] + 1

for some

n

and [·] is the Iverson bracket.

Sequences

A007955 Divisorial of

n

: product of numbers up to

n

which are divisors/factors of

n, n   ≥   1

.

{1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, 225, 1024, 17, 5832, 19, 8000, 441, 484, 23, 331776, 125, 676, 729, 21952, 29, 810000, 31, 32768, 1089, 1156, 1225, 10077696, 37, 1444, 1521, 2560000, 41, ...}

A?????? Almost divisorial primes: Primes

p

such that

p = (   ∏ d ∣n d ∣n d )  −  1

for some

n

(ordered by increasing

n

).

{2, 7, ...}

A?????? Numbers

n

such that

(   ∏ d ∣n d ∣n d )  −  1

is prime.

{3, 4, ...}

A118370 Divisorial primes (quasi divisorial primes): Primes

p

such that

p = (   ∏ d ∣n d ∣n d )  +  1

for some

n

(ordered by increasing

n

).

{2, 3, 37, 101, 197, 331777, 677, 8503057, 9834497, 5477, 59969537, 8837, 17957, 21317, 562448657, 916636177, 42437, 3208542737, 3782742017, 5006411537, 7676563457, 98597, 106277, 11574317057, ...}

A118369 Numbers

n

such that

(   ∏ d ∣n d ∣n d )  +  1

is prime.

{1, 2, 6, 10, 14, 24, 26, 54, 56, 74, 88, 94, 134, 146, 154, 174, 206, 238, 248, 266, 296, 314, 326, 328, 374, 378, 386, 430, 442, 466, 472, 488, 494, 498, 510, 568, 582, ...}

See also

• Sum of divisors (divisor sum)
• Product of divisors (divisor product)
• Divisibility triangle
• Coprimorial (“phi-torial”) and noncoprimorial (“co-phi-torial”)

External links

• Weisstein, Eric W., Divisor Product, from MathWorld—A Wolfram Web Resource.