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# Divisorial

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The **divisorial** of is the product of divisors of , i.e. the product of all positive integers up to which divide .

By analogy with the term phi-torial for the coprimorial of (a product involving numbers,) the divisorial of might be called the tau-torial since the product involves numbers, where is the number of divisors of .

## Contents |

## Formulae

Thus the **divisorial** of is

where is the number of divisors of and is the Iverson bracket.

The following identity holds

where is the number of divisors of .

Note that is always a square since is odd iff is a square.

From the canonical prime factorization of

the canonical prime factorization of is

with

## Properties

All terms of this sequence occur only once.^{[1]}

## 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 such that

for some and is the Iverson bracket.

### Quasi divisorial primes

The quasi divisorial primes are primes such that

for some and is the Iverson bracket.

## Sequences

Divisorial of : product of numbers up to which are divisors/factors of (A007955) gives

- {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, ...}

Almost divisorial primes: Primes such that for some (ordered by increasing .)

- {2, 7, ...}

Numbers such that is prime.

- {3, 4, ...}

Divisorial primes (quasi divisorial primes): Primes such that for some (ordered by increasing .) (A118370)

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

Numbers such that is prime. (A118369)

- {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)
- Divisibility triangle
- Coprimorial (phi-torial) and noncoprimorial (co-phi-torial)

## Notes

- ↑ T. D. Noe, The Divisor Product is Unique.

## External links

- Weisstein, Eric W., Divisor Product, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DivisorProduct.html]