A085157 -id:A085157 - cp's OEIS frontend

A114790 Cumulative product of quintuple factorial A085157.

1, 1, 2, 6, 24, 120, 720, 10080, 241920, 8709120, 435456000, 28740096000, 4828336128000, 1506440871936000, 759246199455744000, 569434649591808000000, 601322989968949248000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!!!!! * 2!!!!! * 3!!!!! * 4!!!!! * 5!!!!! * 6!!!!! * 7!!!!! * 8!!!!! * 9!!!!! * 10!!!!! = 1 * 2 * 3 * 4 * 5 * 6 * 14 * 24 * 36 * 50 = 435456000 = 2^11 * 3^5 * 5^3 * 7.

G. C. Greubel, Table of n, a(n) for n = 0..84
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000178, A006882, A007662, A085157, A114347.

b:= function(n)
    if n<1 then return 1;
    else return n*b(n-5);
    fi;
  end;
List([0..20], n-> Product([0..n], j-> b(j)) ); # G. C. Greubel, Aug 21 2019

b:= func< n | n eq 0 select 1 else (n lt 6) select n else n*Self(n-5) >;
[(&*[b(j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 21 2019

b:= n-> `if`(n < 1, 1, n*b(n-5)); a:= n-> product(b(j), j = 0..n); seq(a(n), n = 0..20); # G. C. Greubel, Aug 21 2019

b[n_]:= If[n<1, 1, n*b[n-5]]; a[n_]:= Product[b[j], {j,0,n}]; Table[a[n], {n,0,20}] (* G. C. Greubel, Aug 21 2019 *)

b(n)=if(n<1, 1, n*b(n-5));
vector(20, n, n--; prod(j=0,n, b(j)) ) \\ G. C. Greubel, Aug 21 2019

@CachedFunction
def b(n):
    if (n<1): return 1
    else: return n*b(n-5)
[product(b(j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 21 2019

a(n) = Product_{j=0..n} A085157(j).
a(n) = n!!!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2.
a(n) = n*(n-5)!!!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2.

A114805 Cumulative sum of quintuple factorial numbers n!!!!! (A085157).

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 36, 60, 96, 146, 212, 380, 692, 1196, 1946, 3002, 5858, 11474, 21050, 36050, 58226, 121058, 250226, 480050, 855050, 1431626, 3128090, 6744794, 13409690, 24659690, 42533546, 96820394, 216171626, 442778090, 836528090

Offset: 0

Jonathan Vos Post, Feb 18 2006

a(1) = 2 is prime; a(3) = 7 is prime; a(4) = 11 is prime; and there are no more primes in the sequence. Semiprime values are: a(2) = 4 = 2^2, a(6) = 22, a(10) = 146 = 2 * 73, a(18) = 11474 = 2 * 5737, a(23) = 250226 = 2 * 125113.

			a(10) = 0!5 + 1!5 + 2!5 + 3!5 + 4!5 + 5!5 + 6!5 + 7!5 + 8!5 + 9!5 + 10!5 =
1 + 1 + 2 + 3 + 4 + 5 + 6 + 14 + 24 + 36 + 50 = 146 = 2 * 73.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A007662, A085157, A114347.

b:= function(n)
    if n<1 then return 1;
    else return n*b(n-5);
    fi;
  end;
List([0..40], n-> Sum([0..n], j-> b(j)) ); # G. C. Greubel, Aug 21 2019

b:= func< n | n eq 0 select 1 else (n lt 6) select n else n*Self(n-5) >;
[(&+[b(j): j in [0..n]]): n in [0..40]]; // G. C. Greubel, Aug 21 2019

b:= n-> `if`(n < 1, 1, n*b(n-5)); a:= n-> sum(b(j), j = 0..n); seq(a(n), n = 0..40); # G. C. Greubel, Aug 21 2019

f5[0]=1; f5[n_]:= f5[n]= If[n<=6, n, n f5[n-5]]; Accumulate[f5/@Range[0, 35]] (* Giovanni Resta, Jun 15 2016 *)

b(n)=if(n<1, 1, n*b(n-5));
vector(40, n, n--; sum(j=0,n, b(j)) ) \\ G. C. Greubel, Aug 21 2019

@CachedFunction
def b(n):
    if (n<1): return 1
    else: return n*b(n-5)
[sum(b(j) for j in (0..n)) for n in (0..40)] # G. C. Greubel, Aug 21 2019

a(n) = Sum_{j=0..n} j!5.
a(n) = Sum_{j=0..n} j!!!!!.
a(n) = Sum_{j=0..n} A085157(j).

A288718 Primes of the form k!5+1, where k!5 is the quintuple factorial number (A085157).

Original entry on oeis.org

2, 3, 5, 7, 37, 67, 313, 751, 2857, 129169, 576577, 17873857, 54286849, 393750001, 643458817, 19053977918977, 206180145819649, 11716249122484566383298871297, 177636555893291390456871518209, 49055724379818682505120501943238657

Offset: 1

Robert Price, Jun 13 2017

nonn

Robert Price, Table of n, a(n) for n = 1..28
Henri& Renaud Lifchitz, PRP Records.Search for n!5+1.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A085148.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 5] + 1, {i, 0, 100}], PrimeQ[#]&]

A289758 Primes of the form k!5-1, where k!5 is the quintuple factorial number (A085157).

Original entry on oeis.org

2, 3, 5, 13, 23, 167, 311, 503, 1696463, 3616703, 119351231, 393749999, 388856692223, 35437499999999, 728640635326636031, 3885182313590882303, 19837740893195045044223, 5126863427472785697108393983, 140901732869280543971697196500573487103

Offset: 1

Robert Price, Jul 11 2017

nonn

Robert Price, Table of n, a(n) for n = 1..27
Henri & Renaud Lifchitz, PRP Records.Search for n!5-1.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A085149.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 5] - 1, {i, 2, 100}], PrimeQ[#]&]

A007661 Triple factorial numbers a(n) = n!!!, defined by a(n) = n*a(n-3), a(0) = a(1) = 1, a(2) = 2. Sometimes written n!3.

Original entry on oeis.org

1, 1, 2, 3, 4, 10, 18, 28, 80, 162, 280, 880, 1944, 3640, 12320, 29160, 58240, 209440, 524880, 1106560, 4188800, 11022480, 24344320, 96342400, 264539520, 608608000, 2504902400, 7142567040, 17041024000, 72642169600, 214277011200, 528271744000, 2324549427200

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

The triple factorial of a positive integer n is the product of the positive integers <= n that have the same residue modulo 3 as n. - Peter Luschny, Jun 23 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

T. D. Noe, Table of n, a(n) for n = 0..200
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Eric Weisstein's World of Mathematics, Multifactorial.

Union of A007559, A008544 and A032031.
Cf. A000142, A006882 (= A001147 union A000165), A007662 (= union of A007696, A001813, A008545 and A047053), A085157, A085158.
Cf. A008585, A016777, A016789, A161474, A288055.

a:= function(n)
    if n<3 then return Fibonacci(n+1);
    else return n*a(n-3);
    fi;
  end;
List([0..30], n-> a(n) ); # G. C. Greubel, Aug 21 2019

a007661 n k = a007661_list !! n
a007661_list = 1 : 1 : 2 : zipWith (*) a007661_list [3..]
-- Reinhard Zumkeller, Sep 20 2013

I:=[1,1,2];[n le 3 select I[n] else (n-1)*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Nov 27 2015

A007661 := n -> mul(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A007661(n), n = 0 .. 29);  # Peter Luschny, Jun 23 2011

multiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*multiFactorial[n - k, k]]]; Array[ multiFactorial[#, 3] &, 30, 0] (* Robert G. Wilson v, Apr 23 2011 *)
RecurrenceTable[{a[0]==a[1]==1,a[2]==2,a[n]==n*a[n-3]},a,{n,30}] (* Harvey P. Dale, May 17 2012 *)
Table[With[{q = Quotient[n + 2, 3]}, 3^q q! Binomial[n/3, q]], {n, 0, 30}] (* Jan Mangaldan, Mar 21 2013 *)
a[ n_] := With[{m = Mod[n, 3, 1], q = 1 + Quotient[n, 3, 1]}, If[n < 0, 0, 3^q Pochhammer[m/3, q]]]; (* Michael Somos, Feb 24 2019 *)
Table[Times@@Range[n,1,-3],{n,0,30}] (* Harvey P. Dale, Sep 12 2020 *)

A007661(n,d=3)=prod(i=0,(n-1)\d,n-d*i) \\ M. F. Hasler, Feb 16 2008

def a(n):
    if (n<3): return fibonacci(n+1)
    else: return n*a(n-3)
[a(n) for n in (0..30)] # G. C. Greubel, Aug 21 2019

a(n) = Product_{i=0..floor((n-1)/3)} (n-3*i). - M. F. Hasler, Feb 16 2008
a(n) ~ c * n^(n/3+1/2)/exp(n/3), where c = sqrt(2*Pi/3) if n=3*k, c = sqrt(2*Pi)*3^(1/6) / Gamma(1/3) if n=3*k+1, c = sqrt(2*Pi)*3^(-1/6) / Gamma(2/3) if n=3*k+2. - Vaclav Kotesovec, Jul 29 2013
a(3*n) = A032031(n); a(3*n+1) = A007559(n+1); a(3*n+2) = A008544(n+1). - Reinhard Zumkeller, Sep 20 2013
0 = a(n)*(a(n+1) -a(n+4)) +a(n+1)*a(n+3) for all n>=0. - Michael Somos, Feb 24 2019
Sum_{n>=0} 1/a(n) = A288055. - Amiram Eldar, Nov 10 2020

A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 16, 27, 40, 55, 72, 91, 224, 405, 640, 935, 1296, 1729, 4480, 8505, 14080, 21505, 31104, 43225, 116480, 229635, 394240, 623645, 933120, 1339975, 3727360, 7577955, 13404160, 21827575, 33592320, 49579075, 141639680

Offset: 0

Hugo Pfoertner, Jun 21 2003

nonn

The term "Sextuple factorial numbers" is also used for the sequences A008542, A008543, A011781, A047058, A047657, A049308, which have a different definition. The definition given here is the one commonly used.

			a(14) = 224 because 14*a(14-6) = 14*a(8) = 14*16 = 224.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!:A085157, 6-factorial primes: n!!!!!!+1:A085150, n!!!!!!-1:A051592.

Cf. A288093.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-6);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 21 2019

b:=func< n | n le 6 select n else n*Self(n-6) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 21 2019

a:= n-> `if`(n<1, 1, n*a(n-6)); seq(a(n), n=0..40); # G. C. Greubel, Aug 21 2019

Table[Times@@Range[n,1,-6],{n,0,40}] (* Harvey P. Dale, Aug 10 2019 *)

a(n)=if(n<1, 1, n*a(n-6));
vector(40, n, n--; a(n) ) \\ G. C. Greubel, Aug 21 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-6)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 21 2019

a(n)=1 for n < 1, otherwise a(n) = n*a(n-6).

Sum_{n>=0} 1/a(n) = A288093. - Amiram Eldar, Nov 10 2020

A008542 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+1).

Original entry on oeis.org

1, 1, 7, 91, 1729, 43225, 1339975, 49579075, 2131900225, 104463111025, 5745471106375, 350473737488875, 23481740411754625, 1714167050058087625, 135419196954588922375, 11510631741140058401875, 1047467488443745314570625, 101604346379043295513350625

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

a(n), n>=1, enumerates increasing heptic (7-ary) trees with n vertices. - Wolfdieter Lang, Sep 14 2007; see a D. Callan comment on A007559 (number of increasing quarterny trees).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for sequences related to factorial numbers.

Cf. A085158, A034689, A034723, A034724, A034787, A034788, A004993, A047058, A047657, A051151.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A045754 (and A084947, A114799), A045755.

List([0..20], n-> Product([0..n-1], k-> (6*k+1) )); # G. C. Greubel, Aug 17 2019

[1] cat [(&*[(6*k+1): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 17 2019

a := n -> mul(6*k+1, k=0..n-1);
G(x):=(1-6*x)^(-1/6): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..15); # Zerinvary Lajos, Apr 03 2009

Table[Product[(6*k+1), {k,0,n-1}], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008, modified by G. C. Greubel, Aug 17 2019 *)
FoldList[Times, 1, 6Range[0, 20] + 1] (* Vincenzo Librandi, Jun 10 2013 *)
Table[6^n*Pochhammer[1/6, n], {n,0,20}] (* G. C. Greubel, Aug 17 2019 *)

a(n)=prod(k=1,n-1,6*k+1) \\ Charles R Greathouse IV, Jul 19 2011

[product((6*k+1) for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 17 2019

E.g.f.: (1-6*x)^(-1/6).
a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(1/6)^-1*n^(-1/3)*6^n*e^-n*n^n*{1 + 1/72*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Sum_{k=0..n} (-6)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1+x/(1-7x/(1-6x/(1-13x/(1-12x/(1-19x/(1-18x/(1-25x/(1-24x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-5)^n*Sum_{k=0..n} (6/5)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0) where Q(k) = 1 - x*(6*k+1)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = A085158(6*n-5). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-6*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/6^5)^(1/6)*(Gamma(1/6) - Gamma(1/6, 1/6)). - Amiram Eldar, Dec 18 2022

A045754 7-fold factorials: a(n) = Product_{k=0..n-1} (7*k+1).

Original entry on oeis.org

1, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 337767408000, 21617114112000, 1534815101952000, 119715577952256000, 10175824125941760000, 936175819586641920000, 92681406139077550080000, 9824229050742220308480000, 1110137882733870894858240000

Offset: 0

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 0..338
Index entries for sequences related to factorial numbers.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A008542 (and A085158), A045755.
See also A113134.
Unsigned row sums of triangle A051186 (scaled Stirling1).
First column of triangle A132056 (S2(8)).
Cf. A114799, A084947, A144739, A144827, A147585, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+1) ); # G. C. Greubel, Aug 21 2019

[1] cat [&*[7*j+1: j in [0..n-1]]: n in [1..20]]; // G. C. Greubel, Aug 21 2019

f := n->product( (7*k+1), k=0..(n-1));
G(x):=(1-7*x)^(-1/7): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

FoldList[Times, 1, 7Range[0, 20] + 1] (* Harvey P. Dale, Jan 21 2013 *)

PARI
```
a(n)=prod(k=0,n-1,7*k+1)
```

[7^n*rising_factorial(1/7, n) for n in (0..20)] # G. C. Greubel, Aug 21 2019

a(n) = Sum_{k=0..n} (-7)^(n-k)*A048994(n, k), where A048994 = Stirling-1 numbers.
E.g.f.: (1-7*x)^(-1/7).
G.f.: 1/(1-x/(1-7*x/(1-8*x/(1-14*x/(1-15*x/(1-21*x/(1-22*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-6)^n*Sum_{k=0..n} (7/6)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/G(0), where G(k)= 1 - x*(7*k+1)/(1 - x*(7*k+7)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k+1)/(x*(7*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
a(n) = 7^n * Gamma(n + 1/7) / Gamma(1/7). - Artur Jasinski, Aug 23 2016
a(n) = A114799(7n-6). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-7*n+6)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/7^6)^(1/7)*(Gamma(1/7) - Gamma(1/7, 1/7)). - Amiram Eldar, Dec 19 2022

Additional comments from Philippe Deléham and Paul D. Hanna, Oct 29 2005
Edited by N. J. A. Sloane, Oct 16 2008 at the suggestion of M. F. Hasler, Oct 14 2008
Corrected by Zerinvary Lajos, Apr 03 2009

A114800 Octuple factorial, 8-factorial, n!8, n!!!!!!!!.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 33, 48, 65, 84, 105, 128, 153, 360, 627, 960, 1365, 1848, 2415, 3072, 3825, 9360, 16929, 26880, 39585, 55440, 74865, 98304, 126225, 318240, 592515, 967680, 1464645, 2106720, 2919735, 3932160, 5175225, 13366080

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 10 * a(10-8) = 10 * a(2) = 10 * 2 = 20.
a(20) = 20 * a(20-8) = 20 * a(12) = 20 * (12*a(12-8)) = 20 * 12 * a(4) = 20 * 12 * 4 = 960.
a(30) = 30 * a(30-8) = 30 * a(22) = 30 * (22*a(22-8)) = 30 * 22 * a(14) = 30 * 22 * (14*a(14-8)) = 30 * 22 * 14 * 6 = 55440.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142, A006882, A007661, A007662, A085157, A085158, A288095.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-8);
    fi;
  end;
List([0..50], n-> a(n) ); # G. C. Greubel, Aug 21 2019

b:=func< n | n le 8 select n else n*Self(n-8) >;
[1] cat [b(n): n in [1..50]]; // G. C. Greubel, Aug 21 2019

A114800 := proc(n)
    option remember;
    if n < 1 then
        1;
    else
        n*procname(n-8) ;
    end if;
end proc:
seq(A114800(n),n=0..40) ; # R. J. Mathar, Jun 23 2014

Table[Times@@Range[n,1,-8],{n,0,50}] (* Harvey P. Dale, Feb 17 2018 *)

a(n)=if(n<1, 1, n*a(n-8));
vector(50, n, n--; a(n) ) \\ G. C. Greubel, Aug 21 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-8)
[a(n) for n in (0..50)] # G. C. Greubel, Aug 21 2019

a(n) = 1 for n<1, else a(n) = n*a(n-8).

Sum_{n>=0} 1/a(n) = A288095. - Amiram Eldar, Nov 10 2020

A085148 Numbers k such that k!!!!! + 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 11, 13, 15, 17, 23, 26, 31, 32, 35, 36, 49, 52, 89, 92, 106, 120, 141, 149, 173, 201, 280, 289, 353, 455, 483, 499, 543, 811, 866, 1010, 1126, 1557, 2358, 2411, 2435, 2485, 2491, 2772, 2851, 2937, 2996, 3642, 3777, 4123, 4642, 5566, 6416, 9202, 9382

Offset: 1

Hugo Pfoertner, Jun 23 2003

The search for multifactorial primes started by Ray Ballinger is now continued by a team of volunteers on the website of Ken Davis (see link).

Robert Price, Table of n, a(n) for n = 1..64
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!5+1.

Cf. A085157 (quintuple factorials), A085149.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

a(56)-a(64), with a(62)-(64) from the Ken Davis link, added to b-file by Robert Price, Sep 23 2012

cp's OEIS Frontend

A114790 Cumulative product of quintuple factorial A085157.

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A114805 Cumulative sum of quintuple factorial numbers n!!!!! (A085157).

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A288718 Primes of the form k!5+1, where k!5 is the quintuple factorial number (A085157).

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A289758 Primes of the form k!5-1, where k!5 is the quintuple factorial number (A085157).

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A007661 Triple factorial numbers a(n) = n!!!, defined by a(n) = n*a(n-3), a(0) = a(1) = 1, a(2) = 2. Sometimes written n!3.

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GAP

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

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GAP

Magma

Maple

Mathematica

PARI

Sage