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Divisorial

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

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

Contents

Formulae

Thus the divisorial of \scriptstyle n is

\tau_{_{_{!}}}(n) = \Pi_{\tau}(n) \equiv \prod_{\stackrel{i = 1}{i|n}}^{n} i = \prod_{i = 1}^{n} i^{[i|n]} = \prod_{i = 1}^{n} i^{[n \bmod i=\, 0]}

where \scriptstyle \tau(n) is the number of divisors of \scriptstyle n and \scriptstyle [\cdot] is the Iverson bracket.

The following identity holds

\Pi_{\tau}(n) = n^{\frac{\tau(n)}{2}} = \sqrt{n^{\tau(n)}}

where \scriptstyle \tau(n) is the number of divisors of \scriptstyle n.

Note that \scriptstyle n^{\tau(n)} is always a square since \scriptstyle \tau(n) is odd iff \scriptstyle n is a square.

From the canonical prime factorization of \scriptstyle n

n = \prod_{i = 1}^{\omega(n)} {p_i}^{\alpha_i}

the canonical prime factorization of \scriptstyle \tau_{_{_{!}}}(n) is

\tau_{_{_{!}}}(n) = \prod_{i = 1}^{\omega(n)} {p_i}^{\beta_i},

with

\beta_i = \frac{\alpha_i}{2} \tau(n) = \frac{\alpha_i}{2} \prod_{j = 1}^{\omega(n)} (\alpha_j+1).

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 \scriptstyle p such that

p = \Bigg( \prod_{d|n} d \Bigg) - 1 = \Bigg( \prod_{i = 1}^{n} d^{[d|n]} \Bigg) - 1

for some \scriptstyle n and \scriptstyle [\cdot] is the Iverson bracket.

Quasi divisorial primes

The quasi divisorial primes are primes \scriptstyle p such that

p = \Bigg( \prod_{d|n} d \Bigg) + 1 = \Bigg( \prod_{i = 1}^{n} d^{[d|n]} \Bigg) + 1

for some \scriptstyle n and \scriptstyle [\cdot] is the Iverson bracket.

Sequences

Divisorial of \scriptstyle n: product of numbers up to \scriptstyle n which are divisors/factors of \scriptstyle n,\ n\ge\, 1, (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 \scriptstyle p such that \scriptstyle p=\, \big( \prod_{d|n} d \big) - 1 for some \scriptstyle n (ordered by increasing \scriptstyle n.)

{2, 7, ...}

Numbers \scriptstyle n such that \scriptstyle \big( \prod_{d|n} d \big) - 1 is prime.

{3, 4, ...}

Divisorial primes (quasi divisorial primes): Primes \scriptstyle p such that \scriptstyle p=\, \big( \prod_{d|n} d \big) + 1 for some \scriptstyle n (ordered by increasing \scriptstyle n.) (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 \scriptstyle n such that \scriptstyle \big( \prod_{d|n} d \big) + 1 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

  1. T. D. Noe, The Divisor Product is Unique.

External links

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