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The superfactorial is a factorial-based function with differing definitions.[1]

## Pickover Edit

Clifford A. Pickover defines superfactorial as \(n\\$ = {}^{(n!)}(n!) = \underbrace{n!^{n!^{n!^{.^{.^.}}}}}_{n!}\) (the factorial of n tetrated to itself or equivalently the factorial of n pentated to 2) in his book Keys to Infinity.

The above is also equal to Template:Arrows or Template:Arrows in up-arrow notation

Using Hypercalc, Wolfram Alpha and bcalc, some values of Pickover's superfactorial are described below:

• 1\$ = 1
• 2\$ = 4
• 3\$ = 1010101036305.315801918918... = 4₧(36305.315801918918)
• 4\$ = 23₧(33.2650153614572)
• 5\$ = 119₧(249.81964187494424)
• 6\$ = 719₧(2057.73535821096)
• 7\$ = 5039₧(18660.818390604654)
• 8\$ = 40319₧(185695.25078372072)
• 9\$ = 362879₧(2017527.5544266126)
• 10\$ = 3628799₧(23804068.910591464)
• 11\$ = 39916799₧(303413813.4465399)
• 12\$ = 479001599₧(4157895295.2732654)
• 13\$ = 6227020799₧(60989187252.17885)
• 14\$ = 87178291199₧(953766105164.5106)
• 15\$ = 1307674367999₧(1.584443597135*1013)
• ...

## Sloane and Plouffe Edit

Neil J.A. Sloane and Simon Plouffe define superfactorial as n\$ = i! = 1!*2!*3!*4!*...*n!, the product of the first n factorials. The first few values of n\$ for n = 1, 2, 3, \ldots\) are 1, 1, 2, 12, 288, 34,560, 24,883,200, 125,411,328,000, 5,056,584,744,960,000, 1,834,933,472,251,084,800,000, 6,658,606,584,104,736,522,240,000,000, 26,579,026,7296,391,946,810,949,632,000,000,000, 127,313,963,299,399,416,749,559,771,247,411,200,000,000,000, ... (OEIS A000178).

## Sources Edit

1. Superfactorial from Wolfram MathWorld
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