A046882 -id:A046882 - cp's OEIS frontend

A165710 Partial sums of A046882.

1, 2, 6, 46662, 1333735776850284124449081472890438

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

nonn

Accumulate[Table[(n!)^n!,{n,0,5}]]  (* Harvey P. Dale, Apr 01 2011 *)

a(n) = Sum_{k=0..n} (k!)^(k!) = 1 + A165711(n).

Definition simplified by R. J. Mathar, Sep 25 2009

A062961 Number of divisors of n!^n! (A046882).

Original entry on oeis.org

1, 1, 3, 49, 1825, 5285401, 2993246641, 5164747323361921, 37003141448637678721, 485526598340984090835841, 11097683374342583428355232001, 6485725972943960604540744581217509068801

Offset: 0

Jason Earls, Jul 22 2001

nonn

PARI
```
for(n=0,22,print(numdiv(n!^n!)))
```

More terms from Vladeta Jovovic, Jul 26 2001

A165812 Ultraprimorials: a(n) = p#^p#.

Original entry on oeis.org

1, 4, 46656, 205891132094649000000000000000000000000000000

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Sep 28 2009

The n-th Primorial raised to its own power.

Terms from a(4) on have 488, 7770, 134461 and 2913994 digits (A195188) and are too large to be listed explicitly.

Eric Weisstein's World of Mathematics, Ultraprimorial

Cf. A002110, A046882

P:=proc(i) local a,n; a:=1; print(1); for n from 1 by 1 to i do a:=a*ithprime(n); print(a^a); od; end: P(10);

a(n) = A000312(A002110(n)) . [R. J. Mathar, Sep 29 2009]

Slightly edited by R. J. Mathar, Sep 29 2009

A100685 Powers of factorials A000142.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 24, 32, 36, 64, 120, 128, 216, 256, 512, 576, 720, 1024, 1296, 2048, 4096, 5040, 7776, 8192, 13824, 14400, 16384, 32768, 40320, 46656, 65536, 131072, 262144, 279936, 331776, 362880, 518400, 524288, 1048576, 1679616, 1728000

Offset: 1

Kyle Schalm and Jonathan Sondow, Dec 08 2004

Subsequence of A001013. Supersequence of A036740 without its first term.

Supersequence also of A046882 and A055209 without their first terms. - Jonathan Sondow and Robert G. Wilson v, Dec 19 2004

			24 = (4!)^1 and 36 = (3!)^2.

G. C. Greubel, Table of n, a(n) for n = 1..9660
Index entries for sequences related to factorial numbers.

Cf. A000142, A001013, A036740, A331373.
Cf. also A046882 and A055209.
Subsequences: A000079, A000400, A009968.

With[{ln = Log[10!]}, Table[With[{f = m!}, Table[f^j, {j, 0, Floor[ln/Log[f]]}]], {m, 2, 10}]] //Flatten //Union

Sum_{n>=1} 1/a(n) = 1 + A331373. - Amiram Eldar, Nov 21 2021

A195529 Ultracompositorial: Compositorials raised to the power of themselves.

Original entry on oeis.org

1, 256, 1333735776850284124449081472843776

Offset: 0

Kausthub Gudipati, Sep 21 2011

nonn

Next term (192^192) has 439 digits.

Amiram Eldar, Table of n, a(n) for n = 0..3

Cf. A000312, A036691, A046882, A165812.

(* First run the program for A036691 *) Table[A036691[[n]]^A036691[[n]], {n, 4}] (* Alonso del Arte, Sep 21 2011 *)
(#^#) & /@ FoldList[Times, 1, Select[Range[6], CompositeQ]] (* Amiram Eldar, Jul 20 2025 *)

a(n) = A000312(A036691(n)). - Amiram Eldar, Jul 20 2025

a(2) = A114993(2). - R. J. Mathar, Aug 22 2025

a(2) corrected by Franklin T. Adams-Watters, Sep 21 2011

A260614 Product of the first n ultrafactorials ((n!)^(n!)).

Original entry on oeis.org

1, 1, 4, 186624, 248907105618907424441185380787996852224

Offset: 0

Matthew Campbell, Aug 13 2015

a(6) is about 10^2345.

This could be called a superultrafactorial by analogy with factorial and superfactorial.

			a(3) = (1!)^(1!) * (2!)^(2!) * (3!)^(3!) = 1^1 * 2^2 * 6^6 = 1 * 4 * 46656 = 186624.

Matthew Campbell, Table of n, a(n) for n = 0..5
Index to divisibility sequences

Cf. A000178, A046882.

[Factorial(n)^Factorial(n): n in [0..5]]; // Vincenzo Librandi, Aug 29 2015

Table[Product[(m!)^(m!), {m, 0, n}], {n, 0, 5}]

a(n)=prod(k=2,n,k!^k!) \\ Charles R Greathouse IV, Aug 14 2015

a(n) = Product_{k=1..n} A046882(k).

A275587 a(1) = 0; thereafter a(n) = s^s where s = Sum_{k=1..n-1} a(k).

Original entry on oeis.org

0, 1, 1, 4, 46656

Offset: 1

Rick L. Shepherd, Aug 02 2016

nonn

a(2) = 1 by the convention that 0^0 = 1. a(6) = 46662^46662 = 1471657830...9021753344, which has 217864 decimal digits.

Cf. A275588, A046882.

a[0] = 0; a[1] = 1; a[n_] := a[n] = #^# &@ Sum[a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 5}] (* Michael De Vlieger, Aug 04 2016 *)

a(n) = local(s); if(n==1, 0, if(n>1, s = sum(k=1, n - 1, a(k)); s^s))

A073279 Number of digits of n!^n! (ultrafactorial numbers).

Original entry on oeis.org

1, 1, 1, 5, 34, 250, 2058, 18661, 185695, 2017527, 23804069, 303413813, 4157895295, 60989187252, 953766105164, 15844435971349, 278704524938621, 5175632136205088, 101198102131888304, 2078318050691940125, 44731639502987747576, 1006917859088311808499

Offset: 0

Rick L. Shepherd, Jul 23 2002

Cf. A046882 (ultrafactorial numbers), A000142 (n!), A034886 (digits of n!).

Table[1 + Floor[n!*Log10[n!]], {n, 0, 21}] (* Arkadiusz Wesolowski, Oct 13 2012 *)

for(n=0,8,print1(length(Str(n!^n!)),","))

a(n) = floor(log(n!^n!)/log(10)) + 1.

More terms from Arkadiusz Wesolowski, Jul 04 2012

Terms a(17) to a(21) corrected by Arkadiusz Wesolowski, Oct 13 2012

A110094 Startorial primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 719, 5039, 1451521, 2903041, 5806081, 46448639, 92897281, 371589121, 10032906239, 30098718719, 270888468479, 812665405439, 7313988648961, 21941965946881, 89874292518420479

Offset: 1

Jonathan Vos Post, Sep 04 2005

These are primes of the form A109834 startorials (base 10) +1 or -1. This is by analogy to factorial primes (A002981), superfactorial primes (A073828), hyperfactorial primes, ultrafactorial primes (comment in A046882), subfactorial primes (A100015), double factorial primes (A080778), multifactorial primes (A037083).

Cf. A000040, A002981, A073828, A080778, A037083, A100015, A103317, A046882, A109834.

{a(n)} = {A109834(k)+1 an element of A000040, or A109834(k)-1 an element of A000040, for some k}.

A174212 Ultradoublefactorials: a(n) = (n!!)^(n!!).

Original entry on oeis.org

1, 1, 4, 27, 16777216, 437893890380859375, 500702078263459319174537025249570888246709955377400223021257741084821677152403456

Offset: 0

Paolo P. Lava & Giorgio Balzarotti, Mar 16 2010

The next term (a(8)) has 993 digits. - Harvey P. Dale, Aug 17 2017

			For n=4 the doublefactorial is n!! = 4*2 = 8 and a(n) = n!!^n!! = 8^8 = 16777216.

Cf. A006882, A046882, A134877, A165812.

P:=proc(i) local a,n; for n from 0 by 1 to i do print(doublefactorial(n)^doublefactorial(n)); od; end: P(10);

udf[n_]:=Module[{c=n!!},c^c]; Array[udf,7,0] (* Harvey P. Dale, Aug 17 2017 *)

a(n) = A006882(n)^A006882(n).

Sum_{n>=1} 1/a(n) = A134877. - Amiram Eldar, Nov 11 2020

cp's OEIS Frontend

A165710 Partial sums of A046882.

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A062961 Number of divisors of n!^n! (A046882).

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A165812 Ultraprimorials: a(n) = p#^p#.

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A100685 Powers of factorials A000142.

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A195529 Ultracompositorial: Compositorials raised to the power of themselves.

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A260614 Product of the first n ultrafactorials ((n!)^(n!)).

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A275587 a(1) = 0; thereafter a(n) = s^s where s = Sum_{k=1..n-1} a(k).

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A073279 Number of digits of n!^n! (ultrafactorial numbers).

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