A002982 -id:A002982 - 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

A099351 Numbers k such that 5*k! - 1 is prime.

Original entry on oeis.org

3, 5, 8, 13, 20, 25, 51, 97, 101, 241, 266, 521, 1279, 1750, 2204, 2473, 4193, 5181, 10080

Offset: 1

Brian Kell, Oct 12 2004

a(15) > 1879. - Jinyuan Wang, Feb 04 2020

a(17) > 3500. - Michael S. Branicky, Mar 06 2021

			k = 5 is here because 5*5! - 1 = 599 is prime.

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002982, A076133, A076134, A099350, A180627, A180628, A180629, A180630, A180631.

for n from 0 to 1000 do if isprime(5*n! - 1) then print(n) end if end do;

Select[Range[550],PrimeQ[5#!-1]&] (* Harvey P. Dale, Nov 27 2013 *)

is(n)=ispseudoprime(5*n!-1) \\ Charles R Greathouse IV, Jun 13 2017

from sympy import isprime
from math import factorial
print([k for k in range(300) if isprime(5*factorial(k) - 1)]) # Michael S. Branicky, Mar 05 2021

a(13)-a(14) from Jinyuan Wang, Feb 04 2020
a(15)-a(16) from Michael S. Branicky, Mar 05 2021
a(17)-a(18) from Michael S. Branicky, Apr 03 2023
a(19) from Michael S. Branicky, Jul 12 2024

A156167 Numbers n such that n![7]-1 is prime (where n![7] = A114799(n) = septuple factorial).

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 12, 14, 17, 20, 24, 30, 31, 32, 46, 52, 54, 59, 98, 104, 143, 145, 160, 174, 198, 199, 202, 212, 215, 254, 371, 382, 452, 674, 739, 959, 1249, 1657, 2291, 2553, 2650, 3562, 3727, 3853, 4389, 4604, 5449, 5659, 6026, 6878, 7900, 9564, 10150, 12444, 13321, 22642, 24014, 26598, 27430, 31386, 40707, 43328, 45811

Offset: 1

M. F. Hasler, Feb 10 2009

nonn

a(65) > 50000. - Robert Price, Sep 09 2012

C. Caldwell, H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!7-1.

cf. A156165, A051592, A085149, A085147, A084438, A007749, A002982.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[MultiFactorial[#, 7] - 1] & ] (* Robert Price, Apr 19 2019 *)

mf(n,k=7)=prod(i=0,(n-2)\k,n-i*k)
for( n=1,9999, ispseudoprime(mf(n)-1) & print1(n","))

a(43)-a(64) from Robert Price, Sep 09 2012

A180627 Numbers k such that 6*k! - 1 is prime.

Original entry on oeis.org

0, 1, 2, 5, 8, 42, 318, 326, 1054, 2987, 11243

Offset: 1

Robert G. Wilson v, Sep 13 2010

Tested to 4400. - Robert G. Wilson v, Sep 28 2010

a(11) > 6300. - Jinyuan Wang, Feb 04 2020

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002982, A076133, A076134, A099350, A099351, A180628, A180630, A180630, A180631.

fQ[n_] := PrimeQ[6 n! - 1]; k = 0; lst = {}; While[k < 1501, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst

is(k) = ispseudoprime(6*k!-1); \\ Jinyuan Wang, Feb 04 2020

a(10) from Robert G. Wilson v, Sep 28 2010

a(11) from Michael S. Branicky, Jul 04 2024

A180631 Numbers k such that 10*k! - 1 is prime.

Original entry on oeis.org

2, 3, 4, 33, 55, 95, 110, 148, 170, 612, 1155, 2295, 2473, 4143, 5671

Offset: 1

Robert G. Wilson v, Sep 13 2010

a(16) > 12000. - Michael S. Branicky, Jul 04 2024

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002982, A076133, A076134, A099350, A099351, A180627, A180628, A180630, A180630.

fQ[n_] := PrimeQ[10 n! - 1]; k = 0; lst = {}; While[k < 1501, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst

a(12)-a(14) from Jinyuan Wang, Feb 04 2020

a(15) from Michael S. Branicky, Jul 03 2024

A064145 a(n) = tau(n!-1) or number of divisors of n!-1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 6, 4, 16, 2, 4, 2, 24, 4, 8, 8, 8, 4, 16, 8, 4, 4, 8, 4, 4, 16, 32, 2, 8, 2, 2, 4, 8, 4, 32, 2, 16, 4, 16, 16, 128, 16, 32, 32, 4, 16, 8, 4, 32, 32, 16, 64, 64, 32, 64, 32, 4, 8, 16, 16, 32, 16, 64, 16, 128, 4, 64, 32, 32, 8, 16, 32, 128, 8

Offset: 2

Vladeta Jovovic, Sep 11 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 2..135
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002582, A027423, A002981, A002982, A064144.

Do[ Print[ DivisorSigma[0, n! - 1]], {n, 2, 40} ]
DivisorSigma[0,Range[2,80]!-1] (* Harvey P. Dale, Aug 17 2024 *)

{ f=1; for (n=2, 100, f*=n; if (n>1, a=numdiv(f - 1), a=0); write("b064145.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 09 2009

More terms from Robert G. Wilson v, Oct 04 2001
a(51)-a(76) from Harry J. Smith, Sep 09 2009
Ambiguous term a(1) removed by Max Alekseyev, May 06 2022

A180628 Numbers k such that 7*k! - 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 12, 23, 25, 31, 57, 74, 86, 140, 240, 310, 703, 713, 796, 1028, 1102, 1924, 3469, 3990

Offset: 1

Robert G. Wilson v, Sep 13 2010

a(26) > 12000. - Michael S. Branicky, Jul 07 2024

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002982, A076133, A076134, A099350, A099351, A180627, A180629, A180630, A180631.

fQ[n_] := PrimeQ[7 n! - 1]; k = 0; lst = {}; While[k < 1501, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst
Select[Range[4000],PrimeQ[7#!-1]&] (* Harvey P. Dale, Apr 22 2024 *)

is(k) = ispseudoprime(7*k!-1); \\ Jinyuan Wang, Feb 03 2020

a(23) from Jinyuan Wang, Feb 03 2020

a(24)-a(25) from Michael S. Branicky, Apr 25 2023

A180630 Numbers k such that 9*k! - 1 is prime.

Original entry on oeis.org

2, 3, 12, 15, 16, 25, 30, 38, 59, 82, 114, 168, 172, 175, 213, 229, 251, 302, 311, 554, 2538, 3050, 3363, 12316

Offset: 1

Robert G. Wilson v, Sep 13 2010

a(22) > 2575. - Jinyuan Wang, Feb 03 2020

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002982, A076133, A076134, A099350, A099351, A180627, A180628, A180629, A180631.

fQ[n_] := PrimeQ[9 n! - 1]; k = 0; lst = {}; While[k < 1501, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst

is(k) = ispseudoprime(9*k!-1); \\ Jinyuan Wang, Feb 03 2020

a(21) from Jinyuan Wang, Feb 03 2020
a(22)-a(23) from Michael S. Branicky, Apr 25 2023
a(24) from Michael S. Branicky, Nov 02 2024

A051856 Numbers k such that (k!)^2 + k! + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 76, 2837, 6001, 7076

Offset: 1

Andrew Walker (ajw01(AT)uow.edu.au), Dec 13 1999

			6 is in the sequence because (6!)^2+6!+1=519121 is prime.

H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3) (1987)

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [broken link]
M. Oakes, Re: Gaussian primorial and factorial primes, Primeform, Dec 21 2010 [broken link]
Mike Oakes, Andrew Walker, David Broadhurst, Gaussian primorial and factorial primes, digest of 7 messages in primeform Yahoo group, Dec 20 - Dec 21, 2010.

Cf. A002981, A002982, A046029.

Do[If[PrimeQ[n!^2+n!+1], Print[n]], {n, 600}] (* Farideh Firoozbakht, Jul 12 2003 *)

Edited by R. J. Mathar, Aug 08 2008

a(8)-a(10) from Serge Batalov, Nov 24 2011

A180629 Numbers k such that 8*k! - 1 is prime.

Original entry on oeis.org

0, 1, 3, 4, 8, 33, 121, 177, 190, 276, 473, 484, 924, 937, 1722, 2626, 4077, 4464, 6166

Offset: 1

Robert G. Wilson v, Sep 13 2010

Tested to 4700. - Robert G. Wilson v, Sep 27 2010
Tested to 5127. - Jinyuan Wang, Feb 03 2020
Tested to 12000. - Michael S. Branicky, Jul 11 2024

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002982, A076133, A076134, A099350, A099351, A180627, A180628, A180630, A180631.

fQ[n_] := PrimeQ[8 n! - 1]; k = 0; lst = {}; While[k < 1501, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst

is(k) = ispseudoprime(8*k!-1); \\ Jinyuan Wang, Feb 03 2020

a(15)-a(18) from Robert G. Wilson v, Sep 27 2010

a(19) from Michael S. Branicky, May 27 2023

cp's OEIS Frontend

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

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A099351 Numbers k such that 5*k! - 1 is prime.

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A156167 Numbers n such that n![7]-1 is prime (where n![7] = A114799(n) = septuple factorial).

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A180627 Numbers k such that 6*k! - 1 is prime.

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A180631 Numbers k such that 10*k! - 1 is prime.

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A064145 a(n) = tau(n!-1) or number of divisors of n!-1.

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A180628 Numbers k such that 7*k! - 1 is prime.

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A180630 Numbers k such that 9*k! - 1 is prime.

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