A100085 -id:A100085 - cp's OEIS frontend

A046882 Ultrafactorials: a(n) = n!^n!.

1, 1, 4, 46656, 1333735776850284124449081472843776

Offset: 0

Camillo Lamonaca (Camillo.Lamonaca(AT)dva.gov.au)

nonn

a(5) = 3175 042373 780336 892901 667920 556557 182493 442088 021222 004926 225128 381629 943118 937129 098831 435345 716937 405655 305190 657814 877412 786176 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000. - Jonathan Vos Post, Dec 09 2004

Note that, by analogy with factorial primes, subfactorial primes, superfactorial primes and hyperfactorial primes, if a(n)+1 or a(n)-1 is prime, it should be called an ultrafactorial prime. These begin: a(0)+1 = a(1)+1 = 2, a(2)-1 = 3, a(2)+1 = 5. Are there any more? Note that a(3) = 46657 = 13 * 37 * 97 is a 3-brilliant number. a(3)-5, a(3)-3 and a(3)+5 are semiprime; a(3)-7 and a(3)+7 are primes. - Jonathan Vos Post, Dec 09 2004

Jean-Christophe Pain, Bounds on the p-adic valuation of the factorial, hyperfactorial and superfactorial, arXiv:2408.00353 [math.NT], 2024. See p. 6.
Eric Weisstein's World of Mathematics, Ultrafactorial.

Cf. A002109, A100085.

Cf. A000166, A000178, A002982.

lst={};Do[a=n!^n!;AppendTo[lst, a], {n, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)
#^#&/@(Range[0,5]!) (* Harvey P. Dale, Oct 15 2023 *)

Sum_{n>=1} 1/a(n) = A100085. - Amiram Eldar, Nov 11 2020

A134877 Decimal expansion of Sum_{k>=1} 1/(k!!)^(k!!).

Original entry on oeis.org

1, 2, 8, 7, 0, 3, 7, 0, 9, 6, 6, 4, 1, 6, 8, 1, 8, 1, 4, 7, 1, 1, 3, 2, 0, 2, 9, 7, 5, 5, 8, 2, 0, 4, 2, 5, 9, 0, 4, 2, 1, 6, 9, 5, 5, 3, 4, 0, 2, 2, 2, 0, 8, 1, 0, 2, 6, 7, 9, 8, 7, 6, 5, 9, 2, 6, 9, 0, 9, 9, 7, 3, 0, 0, 3, 5, 1, 8, 6, 6, 5, 4, 1, 9, 3, 0, 7, 3, 6, 8, 0, 1, 0, 8, 8, 0, 8, 3, 9, 3, 7, 7, 8, 7, 5

Offset: 1

Artur Jasinski, Nov 14 2007

			Sum_{k>=1} 1/(k!!)^(k!!) = 1.28703709664168181471132029755820425904216955340222081026798765926909...

Cf. A073009, A100085.

RealDigits[N[Sum[1/n^n, {n, 1, 100}], 200]][[1]]

A134878 Decimal expansion of Sum_{k>=1} 1/(k^2)^(k^2).

Original entry on oeis.org

1, 0, 0, 3, 9, 0, 6, 2, 5, 2, 5, 8, 1, 1, 7, 4, 7, 9, 1, 7, 6, 7, 4, 0, 7, 2, 9, 0, 6, 1, 4, 3, 0, 6, 7, 4, 1, 0, 7, 6, 1, 2, 4, 9, 2, 4, 3, 7, 9, 2, 8, 5, 9, 4, 7, 8, 7, 6, 4, 0, 4, 7, 9, 0, 7, 9, 5, 0, 9, 9, 2, 1, 9, 0, 5, 0, 8, 6, 9, 4, 4, 5, 1, 6, 6, 8, 8, 4, 0, 2, 7, 3, 4, 8, 3, 4, 4, 6, 9, 6, 6, 8, 8, 5

Offset: 1

Artur Jasinski, Nov 14 2007

Sum_{k>=1} 1/(k^2)^(k^2) = 1.003906252581174791767407290614306741076124924379285...

Cf. A073009, A100085, A134877.

RealDigits[N[Sum[1/(n^2)^(n^2), {n, 1, 30}], 200]][[1]]

Equals Sum_{n>=1} 1/A008972(n). - R. J. Mathar, Jul 31 2025

A134879 Decimal expansion of Sum_{k>=1} 1/(k^3)^(k^3).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 5, 9, 6, 0, 4, 6, 4, 4, 7, 7, 5, 3, 9, 0, 6, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 5, 5, 1, 6, 5, 2, 3, 3, 7, 2, 8, 4, 5, 8, 8, 8, 8, 8, 3, 7, 9, 8, 9, 7, 5, 9, 3, 7, 6, 8, 3, 7, 2, 0, 8, 4, 9, 2, 0, 2, 8, 5, 0, 1, 1, 5, 8, 4, 6, 2, 0, 8, 2, 0, 3, 7, 4, 9, 4, 4, 6, 3, 3, 8, 5, 6, 0, 8, 4, 0, 0

Offset: 1

Artur Jasinski, Nov 14 2007

Sum_{k>=1} 1/(k^3)^(k^3) = 1.00000005960464477539062500000000000000225516523372...

Cf. A073009, A100085, A134877, A134878.

RealDigits[N[Sum[1/(n^3)^(n^3), {n, 1, 30}], 200]][[1]]

suminf(n=1,n^(-3*n^3)) \\ Charles R Greathouse IV, Dec 26 2011

A134880 Decimal expansion of Sum_{k>=1} 1/(2^k)^(2^k).

Original entry on oeis.org

2, 5, 3, 9, 0, 6, 3, 0, 9, 6, 0, 4, 6, 4, 4, 7, 7, 5, 4, 4, 4, 8, 3, 5, 1, 0, 8, 6, 2, 4, 2, 7, 5, 2, 2, 1, 7, 0, 0, 3, 7, 2, 6, 4, 0, 0, 4, 4, 1, 8, 1, 3, 1, 3, 3, 3, 7, 0, 7, 2, 6, 6, 4, 5, 8, 5, 4, 1, 1, 9, 7, 7, 3, 3, 5, 5, 9, 0, 7, 7, 9, 3, 6, 0, 9, 7, 6, 6, 9, 0, 4, 0, 1

Offset: 0

Artur Jasinski, Nov 14 2007

			0.25390630960464477544483510862427522170037264004418131333707266458541197733559...

Cf. A073009, A100085, A134877, A134878, A134879.

RealDigits[N[Sum[1/(2^n)^(2^n), {n, 1, 30}], 200]][[1]]

suminf(k=1, 1/(2^k)^(2^k)) \\ Michel Marcus, Jan 15 2021

A134883 Decimal expansion of Sum_{n>=1} 1/(n^n+1).

Original entry on oeis.org

7, 3, 9, 9, 4, 7, 9, 4, 3, 4, 9, 5, 4, 6, 5, 5, 1, 2, 2, 5, 6, 0, 2, 5, 5, 3, 0, 7, 3, 4, 9, 9, 4, 7, 8, 2, 0, 5, 6, 1, 1, 0, 6, 6, 5, 7, 4, 2, 2, 4, 3, 9, 6, 2, 8, 7, 4, 5, 4, 5, 6, 5, 1, 9, 9, 9, 8, 0, 4, 3, 0, 8, 5, 4, 0, 8, 4, 8, 8, 1, 0, 2, 8, 9, 7, 3, 9, 5, 3, 1, 1, 2, 0, 7, 1, 2, 1, 5, 6, 8, 2, 0, 5, 9

Offset: 0

Artur Jasinski, Nov 15 2007

Constant formed from sum of reversed Sierpinski numbers of first kind A014566.

			0.7399479434954655122560255307349947820561106657422439628745456519998...

Cf. A014566, A073009, A100085, A134877, A134878, A134879, A134880, A134881, A134882.

RealDigits[N[Sum[1/(n^n + 1), {n, 1, 150}], 100]][[1]] (* first zero removed *)

A134881 Decimal expansion of Sum_{k>=1} 1/(e^k)^(e^k).

Original entry on oeis.org

0, 6, 5, 9, 8, 8, 4, 1, 7, 7, 4, 3, 3, 4, 3, 7, 9, 1, 7, 5, 6, 5, 6, 2, 3, 8, 6, 7, 2, 4, 1, 0, 7, 7, 9, 7, 4, 3, 8, 1, 4, 4, 4, 3, 9, 3, 4, 1, 2, 1, 3, 1, 0, 2, 6, 2, 8, 0, 5, 4, 3, 6, 6, 5, 5, 9, 9, 9, 8, 5, 2, 0, 7, 6, 6, 0, 7, 1, 5, 7, 1, 2, 7, 8, 5, 1, 1, 2, 0, 0, 8, 1, 9, 4, 3, 6, 0, 7, 7, 0, 2

Offset: 0

Artur Jasinski, Nov 14 2007

			0.06598841774334379175656238672410779743814443934121...

Cf. A073009, A100085, A134877, A134878, A134879, A134890.

RealDigits[N[Sum[1/(E^n)^(E^n), {n, 1, 20}], 200]][[1]] (* first two zeros removed *)

A134882 Decimal expansion of Sum_{x>=1} 1/(Pi^x)^(Pi^x).

Original entry on oeis.org

0, 2, 7, 4, 2, 5, 6, 9, 3, 2, 7, 6, 9, 9, 1, 3, 7, 8, 2, 8, 1, 1, 6, 1, 1, 9, 4, 8, 4, 3, 1, 2, 0, 8, 3, 2, 6, 8, 2, 2, 5, 5, 9, 5, 3, 8, 8, 0, 5, 7, 8, 9, 0, 7, 0, 9, 9, 8, 8, 1, 7, 4, 4, 3, 1, 0, 1, 6, 1, 3, 8, 6, 5, 0, 3, 8, 8, 4, 7, 4, 4, 5, 7, 6, 3, 0, 8, 4, 3, 8, 8, 3, 2, 9, 1, 7, 4, 4, 7, 1, 1

Offset: 0

Artur Jasinski, Nov 14 2007

			0.02742569327699137828116119484312083268225595388057...

Cf. A073009, A100085, A134877, A134878, A134879, A134880, A134881, A134883.

RealDigits[N[Sum[1/(Pi^n)^(Pi^n), {n, 1, 20}], 200]][[1]] (* first two zeros removed *)

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A046882 Ultrafactorials: a(n) = n!^n!.

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A134877 Decimal expansion of Sum_{k>=1} 1/(k!!)^(k!!).

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A134878 Decimal expansion of Sum_{k>=1} 1/(k^2)^(k^2).

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A134879 Decimal expansion of Sum_{k>=1} 1/(k^3)^(k^3).

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A134880 Decimal expansion of Sum_{k>=1} 1/(2^k)^(2^k).

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A134883 Decimal expansion of Sum_{n>=1} 1/(n^n+1).

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A134881 Decimal expansion of Sum_{k>=1} 1/(e^k)^(e^k).

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A134882 Decimal expansion of Sum_{x>=1} 1/(Pi^x)^(Pi^x).

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