Ultraprimorial

This article page is a stub, please help by expanding it.

$U(p_{n})\equiv {p_{n}\#}^{p_{n}\#},n\geq 0.\,$

the 0^th ultraprimorial number being the empty product, defined as the multiplicative identity, i.e. 1.

The ultraprimorial numbers give the sequence (Cf. A165812)

{1, 4, 46656, 205891132094649000000000000000000000000000000, ...}

Higher order ultraprimorial numbers

$U_{0}(p_{n})\equiv 1,\quad n\geq 0.\,$ (empty product, defined as the multiplicative identity, i.e. 1)

$U_{1}(p_{n})\equiv {p_{n}\#},\quad n\geq 0.\,$ (primorial numbers)

$U_{2}(p_{n})\equiv {p_{n}\#}^{p_{n}\#},\quad n\geq 0.\,$ (ultraprimorial numbers)

$U_{3}(p_{n})\equiv {p_{n}\#}^{{p_{n}\#}^{p_{n}\#}},\quad n\geq 0.\,$

$U_{4}(p_{n})\equiv {p_{n}\#}^{{p_{n}\#}^{{p_{n}\#}^{p_{n}\#}}},\quad n\geq 0.\,$

...

Definition for natural numbers (ultraprimorials)

$U(n)\equiv {n\#}^{n\#},n\geq 0.\,$

the ultraprimorial of 0 being the empty product, defined as the multiplicative identity, i.e. 1.

The ultraprimorials give the sequence (Cf. A??????)

{1, 1, 4, 46656, 46656, 205891132094649000000000000000000000000000000, 205891132094649000000000000000000000000000000, ...}

Higher order ultraprimorials

$U_{0}(n)\equiv 1,\quad n\geq 0.\,$ (empty product, defined as the multiplicative identity, i.e. 1)

$U_{1}(n)\equiv {n\#},\quad n\geq 0.\,$ (primorials)

$U_{2}(n)\equiv {n\#}^{n\#},\quad n\geq 0.\,$ (ultraprimorials)

$U_{3}(n)\equiv {n\#}^{{n\#}^{n\#}},\quad n\geq 0.\,$

$U_{4}(n)\equiv {n\#}^{{n\#}^{{n\#}^{n\#}}},\quad n\geq 0.\,$

...

See also

• Primorial
• Ultracompositorial
• Ultrafactorial