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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 .

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

Notes

External links