A002982 -id:A002982 - cp's OEIS frontend

A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 6, 6, 5, 3, 1, 1, 4, 8, 11, 12, 11, 8, 4, 1, 1, 4, 8, 11, 12, 12, 12, 12, 11, 8, 4, 1, 1, 4, 8, 12, 16, 19, 20, 20, 19, 16, 12, 8, 4, 1, 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1

Offset: 0

Gus Wiseman, Aug 03 2020

Row n is row n! of A146291. Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  1
  1  2  2  2  1
  1  3  4  4  3  1
  1  3  5  6  6  5  3  1
  1  4  8 11 12 11  8  4  1
  1  4  8 11 12 12 12 12 11  8  4  1
  1  4  8 12 16 19 20 20 19 16 12  8  4  1
Row n = 6 counts the following divisors:
  1  2   4   8  16   48  144  720
     3   6  12  24   72  240
     5   9  18  36   80  360
        10  20  40  120
        15  30  60  180
            45  90
Row n = 7 counts the following divisors:
  1  2   4    8   16   48   144   720  5040
     3   6   12   24   72   240  1008
     5   9   18   36   80   336  1680
     7  10   20   40  112   360  2520
        14   28   56  120   504
        15   30   60  168   560
        21   42   84  180   840
        35   45   90  252  1260
             63  126  280
             70  140  420
            105  210  630
                 315

A000720 is column k = 1.
A008302 is the version for superprimorials.
A022559 gives row lengths minus one.
A027423 gives row sums.
A146291 is the generalization to non-factorials.
A336499 is the restriction to divisors in A130091.
A000142 lists factorial numbers.
A336415 counts uniform divisors of n!.
Cf. A000005, A001222, A118914, A124010, A181796, A327526, A336420.
Factorial numbers: A002982, A007489, A048656, A054991, A071626, A325272, A325617, A336414, A336415, A336416, A336418.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&]],{n,0,10},{k,0,PrimeOmega[n!]}]

A049433 Numbers k such that k! - (k-1)! - 1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 28, 78, 99, 184, 286, 291, 398, 411, 600, 718, 732, 889, 1963, 2240, 2242, 2533, 8800, 11403, 18335, 20277, 21029

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

nonn

There is no further term up to 1400. - Farideh Firoozbakht, Jul 18 2003

a(25) > 12000. [Donovan Johnson, Dec 18 2009]

			6 is a term since 6! - (6-1)! - 1 = 599 is prime.

Index entries for sequences related to factorial numbers

Cf. A049432, A049985, A090704.

Cf. A001272, A002981, A002982.

a(n) = A090704(n) + 1 = n*n! + 1 = ((n+1)-1)*n! + 1 = (n+1)! - n! + 1 .

More terms from Farideh Firoozbakht, Jul 18 2003
Corrected offset, edited definition and a(19)-a(24) from Donovan Johnson, Dec 18 2009
a(25)-a(27) from Michael S. Branicky, Jun 13 2025

A055489 Largest number x such that sum of divisors of x is n!.

Original entry on oeis.org

5, 23, 95, 719, 5039, 39917, 361657, 3624941, 39904153, 479001599, 6226862869, 87178291199, 1307672080867, 20922780738961, 355687390376431, 6402373545694717, 121645099711277873, 2432902005056589697, 51090942157413850441

Offset: 3

Labos Elemer, Jun 28 2000

nonn

For n = 1, a(1) = 1; for n = 2 there is no solution.

For n in A002982, a(n) = n!-1.

			For n = 6, the 15 solutions are as follows: {264, 270, 280, 354, 376, 406, 418, 435, 459, 478, 537, 623, 649, 667, 719}.

R. K. Guy (1981): Unsolved Problems In Number Theory, B39.

Ray Chandler, Table of n, a(n) for n=3..52
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.

Cf. A000203, A000142, A002982, A055486, A055488, A057637.

a(n) = {fn = n!; x = fn - 1; while(sigma(x) != fn, x--); x;} \\ Michel Marcus, Dec 17 2013

a(n) = Max{x; Sigma[x] = n!} = Max{x; A000203(x) = A000142(n)}
a(n) = A057637(A000142(n)). - Ray Chandler, Jan 15 2009
a(A002982(n)) = A000142(A002982(n)) - 1. - Ray Chandler, Jan 15 2009

More terms from Jud McCranie, Oct 09 2000
a(15) from Donovan Johnson, Aug 31 2008
a(16)-a(19) from Donovan Johnson, Nov 22 2008
a(20)-a(52) from Ray Chandler, Jan 15 2009

A064144 a(n) is the number of divisors of n!+1.

Original entry on oeis.org

2, 2, 2, 3, 3, 4, 3, 4, 8, 4, 2, 6, 4, 4, 8, 32, 8, 64, 4, 4, 8, 8, 12, 4, 4, 4, 2, 4, 8, 32, 16, 16, 32, 4, 32, 64, 2, 4, 16, 128, 2, 8, 16, 8, 8, 8, 16, 4, 32, 32, 64, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 8, 32, 8

Offset: 1

Vladeta Jovovic, Sep 11 2001

nonn

Max Alekseyev, Table of n, a(n) for n = 1..139
William Gerst, A conjecture on the prime factorization of n!+1, arXiv:1809.07360 [math.GM], 2018.
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002583, A027423, A002981, A002982, A064145, A078778, A181764.

Do[ Print[ DivisorSigma[0, n! + 1]], {n, 1, 40} ]

a(n) = numdiv(n! + 1); \\ Harry J. Smith, Sep 09 2009

from math import factorial
from sympy import divisor_count
def A064144(n): return divisor_count(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

a(n) = tau(n!+1).

More terms from Robert G. Wilson v, Oct 04 2001
a(42)-a(64) from Harry J. Smith, Sep 09 2009
Edited by Jon E. Schoenfield, Jun 21 2018

A088054 Factorial primes: primes which are within 1 of a factorial number.

Original entry on oeis.org

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Nov 02 2003

Conjecture: 3 is the intersection of A002981 and A002982.

			3! + 1 = 7; 7! - 1 = 5039.
39916801 is a term because 11! + 1 is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..29
C. Caldwell's The Top Twenty, Factorial Primes.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Wikipedia, Factorial prime.

Cf. A000142, A002981, A002982.

Union of A055490 and A088332.

t = {}; Do[ If[PrimeQ[n! - 1], AppendTo[t, n! - 1]]; If[PrimeQ[n! + 1], AppendTo[t, n! + 1]], {n, 50}]; t (* Robert G. Wilson v *)
Union[Select[Range[50]!-1, PrimeQ], Select[Range[50]!+1, PrimeQ]] (Noe)
fp[n_] := Module[{nf=n!}, Select[{nf-1,nf+1},PrimeQ]]; Flatten[ Table[ fp[i],{i,50}]] (* Harvey P. Dale, Dec 18 2010 *)
Select[Flatten[#+{-1,1}&/@(Range[50]!)],PrimeQ] (* Harvey P. Dale, Apr 08 2019 *)

from itertools import count, islice
from sympy import isprime
def A088054_gen(): # generator of terms
    f = 1
    for k in count(1):
        f *= k
        if isprime(f-1):
            yield f-1
        if isprime(f+1):
            yield f+1
A088054_list = list(islice(A088054_gen(),10)) # Chai Wah Wu, Feb 18 2022

Corrected by Paul Muljadi, Oct 11 2005

More terms from Robert G. Wilson v and T. D. Noe, Oct 12 2005

A103317 Primes p such that p! - 1 is also prime.

Original entry on oeis.org

3, 7, 379, 6917, 208003

Offset: 1

Jonathan Sondow, Jan 31 2005

The members are the primes in A002982 (n! - 1 is prime).

			3 is prime and 3! - 1 = 5 is prime, so 3 is a member.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

Eric Weisstein's World of Mathematics, Factorial Prime
Index entries for sequences related to factorial numbers.

Cf. A002982, A093804.

Select[Prime[Range[100]], PrimeQ[#! - 1] &] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

isok(n) = ispseudoprime(n) && ispseudoprime(n!-1); \\ Jinyuan Wang, Jan 20 2020

a(5)=208003 (found on Jul 27 2016, by Sou Fukui), added by Jinyuan Wang, Jan 20 2020

A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0

Offset: 0

Gus Wiseman, Aug 03 2020

Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  0
  1  2  1  2  1
  1  3  1  3  2  0
  1  3  2  5  3  3  2  1
  1  4  2  7  4  4  3  2  0
  1  4  2  7  4  5  7  7  6  3  2  0
  1  4  2  8  8  9 10 11 11  7  8  5  2  0
  1  4  3 11  8 11 16 16 15 15 15 13  9  6  3  1
  1  5  3 14 10 13 21 21 20 19 21 18 13  9  5  2  0
  1  5  3 14 10 14 25 23 27 24 30 28 28 25 20 16 11  5  2  0
Row n = 7 counts the following divisors:
  1  2  4  8   16  48   144  720   {}
     3  9  12  24  72   360  1008
     5     18  40  80   504
     7     20  56  112
           28
           45
           63

A000720 is column k = 1.
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336414 gives row sums.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.
Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]

A073443 Numbers k such that k! - k - 1 is prime.

Original entry on oeis.org

3, 4, 10, 12, 346

Offset: 1

Rick L. Shepherd, Jul 31 2002

Clearly n <> 2 (mod 3). For n>3, n!-n, n!-n+1, ..., n!-3, n!-2 is a sequence of n-1 consecutive composite numbers. Additional terms are greater than 2000.
a(5) > 7500. - Michael S. Branicky, Mar 04 2021
a(5) > 24000. - Michael S. Branicky, Nov 13 2024

Cf. A073444 (corresponding primes), A002982 (n!-1 is prime), A073308 (n!+n+1 is prime).

Select[Range[3, 346], PrimeQ[#! - # - 1] &] (* Arkadiusz Wesolowski, Jan 04 2012 *)

for(n=3,2000,if(isprime(n!-n-1),print1(n,",")))

from math import factorial
from sympy import isprime
def ok(n): return isprime(factorial(n) - n - 1)
print([m for m in range(3, 500) if ok(m)]) # Michael S. Branicky, Mar 04 2021

Offset corrected by Arkadiusz Wesolowski, Jan 04 2012

A325704 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 1, 7, 1, 5, 1, 25, 2, 4, 1, 2, 1, 13, 13, 121, 1, 7, 1, 721, 3, 49, 1, 5, 1, 5, 61, 5041, 5, 3, 1, 40321, 361, 19, 1, 37, 1, 241, 7, 362881, 1, 9, 1, 4, 2521, 1441, 1, 5, 7, 73, 20161, 3628801, 1, 8, 1, 39916801, 25, 6, 121, 181, 1

Offset: 1

Gus Wiseman, May 18 2019

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the numerator of 1/i_1! + ... + 1/i_k!.

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.

Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325703.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Numerator

A325704(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/(primepi(f[i, 1])!))); }; \\ Antti Karttunen, Nov 17 2019

a(n) = A318573(A325709(n)).

A051855 Numbers n such that (n!)^4+1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 13, 112, 328, 11123

Offset: 1

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

nonn

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [?Broken link]

Cf. A002981, A002982, A046029.

[n: n in [1..300] | IsPrime(Factorial(n)^4+1)]; // Vincenzo Librandi, Aug 15 2013

Select[Range[0, 350], PrimeQ[(#!)^4 + 1]&] (* Vincenzo Librandi, Aug 15 2013 *)

isok(n) = isprime(n!^4 + 1); \\ Michel Marcus, Aug 15 2013

a(9) from Robert Price, Jul 24 2014

Prepended a(1)=0, Robert Price, Sep 01 2014

cp's OEIS Frontend

A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.

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A049433 Numbers k such that k! - (k-1)! - 1 is prime.

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A055489 Largest number x such that sum of divisors of x is n!.

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A064144 a(n) is the number of divisors of n!+1.

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A088054 Factorial primes: primes which are within 1 of a factorial number.

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A103317 Primes p such that p! - 1 is also prime.

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A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

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Mathematica

A073443 Numbers k such that k! - k - 1 is prime.