A085157 -id:A085157 - cp's OEIS frontend

A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

1, 1, 2, 3, 4, 5, 6, 7, 8, 18, 30, 44, 60, 78, 98, 120, 288, 510, 792, 1140, 1560, 2058, 2640, 6624, 12240, 19800, 29640, 42120, 57624, 76560, 198720, 379440, 633600, 978120, 1432080, 2016840, 2756160, 7352640, 14418720, 24710400, 39124800

Offset: 0

Jonathan Vos Post, Feb 18 2006

Many of the terms yield multifactorial primes a(n) + 1, e.g.: a(2) + 1 = 3, a(4) + 1 = 5, a(6) + 1 = 7, a(9) + 1 = 19, a(10) + 1 = 31, a(12) + 1 = 61, a(13) + 1 = 79, a(24) + 1 = 12241, a(25) + 1 = 19801, a(26) + 1 = 29641, a(29) + 1 = 76561, a(31) + 1 = 379441, a(35) + 1 = 2016841, a(36) + 1 = 2756161, ...

Equivalently, product of all positive integers <= n congruent to n (mod 7). - M. F. Hasler, Feb 23 2018

			a(40) = 40 * a(40-7) = 40 * a(33) = 40 * (33*a(26)) = 40 * 33 * (26*a(19)) = 40 * 33 * 26 * (19*a(12)) = 40 * 33 * 26 * 19 * (12*a(5)) = 40 * 33 * 26 * 19 * 12 5 = 39124800.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers

Cf. A045754, A084947, A288094.

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), A085158 (and A008542, A047058, A047657), A045755.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-7);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 20 2019

b:= func< n | (n lt 8) select n else n*Self(n-7) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 20 2019

A114799 := proc(n)
    option remember;
    if n < 1 then
        1;
    else
        n*procname(n-7) ;
    end if;
end proc:
seq(A114799(n),n=0..40) ; # R. J. Mathar, Jun 23 2014
A114799 := n -> product(n-7*k,k=0..(n-1)/7); # M. F. Hasler, Feb 23 2018

a[n_]:= If[n<1, 1, n*a[n-7]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 20 2019 *)

A114799(n,k=7)=prod(j=0,(n-1)\k,n-j*k) \\ M. F. Hasler, Feb 23 2018

def a(n):
    if (n<1): return 1
    else: return n*a(n-7)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 20 2019

a(n) = 1 for n <= 1, else a(n) = n*a(n-7).

Sum_{n>=0} 1/a(n) = A288094. - Amiram Eldar, Nov 10 2020

Edited by M. F. Hasler, Feb 23 2018

A114806 Nonuple factorial, 9-factorial, n!9, n!!!!!!!!!.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 440, 756, 1144, 1610, 2160, 2800, 3536, 4374, 5320, 12760, 22680, 35464, 51520, 71280, 95200, 123760, 157464, 196840, 484880, 884520, 1418560, 2112320, 2993760, 4093600

Offset: 0

Jonathan Vos Post, Feb 19 2006

			a(10) = 10 * a(10-9) = 10 * a(1) = 10 * 1 = 10.
a(20) = 20 * a(20-9) = 20 * a(11) = 20 * (11*a(11-9)) = 20 * 11 * a(2) = 20 * 11 * 2 = 440.
a(30) = 30 * a(30-9) = 30 * a(21) = 30 * (21*a(21-9)) = 30 * 21 * a(12) = 30 * 21 * (12*a(12-9)) = 30 * 21 * 12 * 3 = 22680.

Robert Israel, Table of n, a(n) for n = 0..2955
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142, A006882, A007661, A007662, A085157, A085158, A288096.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-9);
    fi;
  end;
List([0..50], n-> a(n) ); # G. C. Greubel, Aug 21 2019

b:=func< n | n le 9 select n else n*Self(n-9) >;
[1] cat [b(n): n in [1..50]]; // G. C. Greubel, Aug 21 2019

f:= proc(n) option remember;
n*procname(n-9)
end proc:
f(0):= 1: for n from 1 to 8 do f(n):= n od:
map(f, [$0..100]); # Robert Israel, Jun 21 2019

NFactorialM[n_, m_] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Array[ NFactorialM[#, 9] &, 44, 0] (* Robert G. Wilson v, May 10 2011 *)
a[n_]:= a[n]= If[n<1, 1, n*a[n-9]]; Table[a[n], {n,0,50}] (* G. C. Greubel, Aug 21 2019 *)
Table[Times@@Range[n,1,-9],{n,0,50}] (* Harvey P. Dale, Nov 13 2021 *)

a(n)=if(n<1, 1, n*a(n-9));
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-9)
[a(n) for n in (0..50)] # G. C. Greubel, Aug 21 2019

D-finite with recurrence: a(0) = 1, a(n) = n for 1 <= n <= 9, a(n) = n*a(n-9) for n >= 10.
From Robert Israel, Jun 21 2019: (Start)
a(9*m) = 9^m*m!.
a(9*m+k) = 9^m*(9*m+k)*Gamma(m+k/9)/Gamma(k/9) for 1 <= k <= 8. (End)
Sum_{n>=0} 1/a(n) = A288096. - Amiram Eldar, Nov 10 2020

A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.640224467733809734176937236963569060632409516968842599452955763...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A085157 (n!5), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), this sequence (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(105)); (1/5)*Exp(1/5)*(5 + (&+[5^(k/5)*Gamma(k/5, 1/5): k in [1..4]])); // G. C. Greubel, Mar 28 2019

m[5] = (1/5)*E^(1/5)*(5 + 5^(1/5)*(Gamma[1/5] - Gamma[1/5, 1/5]) + 5^(2/5)*(Gamma[2/5] - Gamma[2/5, 1/5]) + 5^(3/5)*(Gamma[3/5] - Gamma[3/5, 1/5]) + 5^(4/5)*(Gamma[4/5] - Gamma[4/5, 1/5])); RealDigits[m[5], 10, 102][[1]]

default(realprecision, 105); (1/5)*exp(1/5)*(5 + sum(k=1,4, 5^(k/5)*(gamma(k/5) - incgam(k/5, 1/5)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/5)*exp(1/5)*(5 + sum(5^(k/5)*(gamma(k/5) - gamma_inc(k/5, 1/5)) for k in (1..4))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288327 Decuple factorial, 10-factorial, n!10, n!!!!!!!!!!.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 39, 56, 75, 96, 119, 144, 171, 200, 231, 528, 897, 1344, 1875, 2496, 3213, 4032, 4959, 6000, 7161, 16896, 29601, 45696, 65625, 89856, 118881, 153216, 193401, 240000, 293601, 709632, 1272843, 2010624, 2953125, 4133376

Offset: 0

Robert Price, Jun 07 2017

			a(13) = 13 * 3 * 1 = 39.

Robert Price, Table of n, a(n) for n = 0..803
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A006852, A007661, A007662, A085157, A085158, A114799, A114800, A114806, A342033.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-10);
    fi;
  end;
List([0..50], n-> a(n) ); # G. C. Greubel, Aug 22 2019

b:=func< n | n le 10 select n else n*Self(n-10) >;
[1] cat [b(n): n in [1..50]]; // G. C. Greubel, Aug 22 2019

a:= n-> `if`(n<1, 1, n*a(n-10)); seq(a(n), n=0..50); # G. C. Greubel, Aug 22 2019

MultiFactorial[n_, k_]:=If[n<1, 1 ,n*MultiFactorial[n-k, k]];
Table[MultiFactorial[i, 10], {i, 0, 100}]
Table[Times@@Range[n,1,-10],{n,0,50}] (* Harvey P. Dale, Aug 11 2019 *)

a(n)=if(n<1, 1, n*a(n-10));
vector(40, n, n--; a(n) ) \\ G. C. Greubel, Aug 22 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-10)
[a(n) for n in (0..50)] # G. C. Greubel, Aug 22 2019

a(n)=1 for n < 1, otherwise a(n) = n*a(n-10).

Sum_{n>=0} 1/a(n) = A342033. - Amiram Eldar, May 23 2022

A085149 Numbers k such that k!!!!! - 1 is prime.

Original entry on oeis.org

3, 4, 6, 7, 8, 12, 13, 14, 27, 28, 33, 35, 44, 50, 62, 64, 74, 88, 114, 140, 142, 242, 248, 262, 270, 284, 395, 473, 582, 600, 634, 707, 805, 882, 907, 1008, 1152, 1243, 1853, 2340, 2410, 2703, 2793, 3033, 3600, 3925, 4124, 4154, 6185, 6367, 7130, 9104, 9992

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..63
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!5-1.

Cf. A085157 (quintuple factorials), A085148.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[MultiFactorial[#, 5] - 1] & ] (* Robert Price, Apr 19 2019 *)

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

a(54)-a(63) from Ken Davis link added to b-file by Robert Price, Sep 23 2012

A129116 Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 3, 24, 1, 2, 3, 8, 120, 1, 2, 3, 4, 15, 720, 1, 2, 3, 4, 10, 48, 5040, 1, 2, 3, 4, 5, 18, 105, 40320, 1, 2, 3, 4, 5, 12, 28, 384, 362880, 1, 2, 3, 4, 5, 6, 21, 80, 945, 3628800, 1, 2, 3, 4, 5, 6, 14, 32, 162, 3840, 39916800, 1, 2, 3, 4, 5, 6, 7, 24, 45, 280, 10395, 479001600

Offset: 1

Jonathan Vos Post, May 24 2007

The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used. This problem exists for the other rows as well. "n!2" = n!!, "n!3" = n!!!, "n!4" = n!!!!, etcetera. Main diagonal is A[n,n] = n!n = n.

Similar to A114423 (with rows and columns exchanged). - Georg Fischer, Nov 02 2021

			Table begins:
  k / A(k,n)
  1 | 1 2 6 24 120 720 5040 40320 362880 3628800 ... = A000142.
  2 | 1 2 3  8  15  48  105   384    945    3840 ... = A006882.
  3 | 1 2 3  4  10  18   28    80    162     280 ... = A007661.
  4 | 1 2 3  4   5  12   21    32     45     120 ... = A007662.
  5 | 1 2 3  4   5   6   14    24     36      50 ... = A085157.
  6 | 1 2 3  4   5   6    7    16     27      40 ... = A085158.

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142 (n!), A006882 (n!!), A007661 (n!!!), A007662(n!4), A085157 (n!5), A085158 (n!6), A114799 (n!7), A114800 (n!8), A114806 (n!9), A288327 (n!10).

Cf. A114423 (transposed).

A:= proc(k,n) option remember; if n >= 1 then n* A(k, n-k) elif n >= 1-k then 1 else 0 fi end: seq(seq(A(1+d-n, n), n=1..d), d=1..16); # Alois P. Heinz, Feb 02 2009

A[k_, n_] := A[k, n] = If[n >= 1, n*A[k, n-k], If[n >= 1-k, 1, 0]]; Table[ A[1+d-n, n], {d, 1, 16}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)

A(k,n) = n!k.

A(k,n) = M(n,k) in A114423. - Georg Fischer, Nov 02 2021

Corrected and extended by Alois P. Heinz, Feb 02 2009

A297707 a(n) = Product_{k=1..n-1} n!k, where n!k is k-tuple factorial of n.

Original entry on oeis.org

1, 2, 18, 768, 90000, 44789760, 30494620800, 121762322841600, 393644011735296000, 5618427494400000000000, 107587910030480590233600000, 5951222311476064581656248320000, 176804782652901880753915871232000000, 69819090744423637487544223697731584000000

Offset: 1

Lechoslaw Ratajczak, Jan 03 2018

nonn

What is the least n > 2 for which a(n) - prevprime(a(n)) is a composite number? If such a number n exists, it is greater than 250.

The least n for which nextprime(a(n)) - a(n) is a composite number is 158.

			a(2) = (2!1) = (2*1) = 2;
a(3) = (3!1)*(3!2) = (3*2*1)*(3*1) = 18;
a(4) = (4!1)*(4!2)*(4!3) = (4*3*2*1)*(4*2)*(4*1) = 768;
a(5) = (5!1)*(5!2)*(5!3)*(5!4) = (5*4*3*2*1)*(5*3*1)*(5*2)*(5*1) = 90000.

Michel Marcus, Table of n, a(n) for n = 1..100

Cf. A000142, A006882, A007661, A007662, A085157, A085158, A114799, A114800.

Cf. A114806, A288327, A006990, A033933.

b:= proc(n, k) option remember; `if`(n<1, 1, n*b(n-k, k)) end:
a:= n-> mul(b(n, k), k=1..n-1):
seq(a(n), n=1..20);  # Alois P. Heinz, Dec 02 2018

Array[(#^(# - 1)) Product[k^DivisorSigma[0, # - k], {k, # - 1}] &, 13] (* Michael De Vlieger, Jan 04 2018 *)

a(n) = (n^(n-1))*prod(k=1, n-1, k^numdiv(n-k)); \\ Michel Marcus, Dec 02 2018

a(n) = Product_{t=1..n-1} (Product_{k=0..floor((n-1)/t)} (n-t*k)).

a(n) = (n^(n-1))*Product_{k=1..n-1} k^tau(n-k).

A114423 Multifactorial array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 2, 1, 1, 24, 3, 2, 1, 1, 120, 8, 3, 2, 1, 1, 720, 15, 4, 3, 2, 1, 1, 5040, 48, 10, 4, 3, 2, 1, 1, 40320, 105, 18, 5, 4, 3, 2, 1, 1, 362880, 384, 28, 12, 5, 4, 3, 2, 1, 1, 3628800, 945, 80, 21, 6, 5, 4, 3, 2, 1, 1, 39916800, 3840, 162, 32, 14, 6, 5, 4, 3, 2, 1, 1

Offset: 0

Jonathan Vos Post, Feb 12 2006

The columns are n!, n!!, n!!!, ... n!k for n >= 0, k >= 1.

			Table M begins:
  n / M(n,k)
  0 |   1   1   1   1   1
  1 |   1   1   1   1   1
  2 |   2   2   2   2   2
  3 |   6   3   3   3   3
  4 |  24   8   4   4   4
  5 | 120  15  10   5   5
  6 | 720  48  18  12   6

Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142 (n!), A006882 (n!!), A007661 (n!!!), A007662(n!4), A085157 (n!5), A085158 (n!6), A114799 (n!7), A114800 (n!8), A114806 (n!9), A288327 (n!10).

Cf. A129116 (transposed).

NFactorialM[n_, m_] := Block[{k = n, p = Max[1, n]},
     While[k > m, k -= m; p *= k]; p];
Table[NFactorialM[n - m + 1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Aug 01 2021, after Robert G. Wilson v in A007662 *)

M(n,k) = n!k.

M(n,k) = A129116(k,n). - Georg Fischer, Nov 02 2021

Edited by Alois P. Heinz, Apr 24 2025

A114796 Cumulative product of sextuple factorial A085158.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 80640, 2177280, 87091200, 4790016000, 344881152000, 31384184832000, 7030057402368000, 2847173247959040000, 1822190878693785600000, 1703748471578689536000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!6 * 2!6 * 3!6 * 4!6 * 5!6 * 6!6 * 7!6 * 8!6 * 9!6 * 10!6
= 1 * 2 * 3 * 4 * 5 * 6 * 7 * 16 * 27 * 40 = 87091200 = 2^11 * 3^5 * 5^2 * 7.
Note that a(10) + 1 = 87091201 is prime, as is a(9) + 1 = 2177281.

G. C. Greubel, Table of n, a(n) for n = 0..91
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000178, A006882, A007662, A085150, A085157, A085158, A114347.

b:= function(n)
    if n<1 then return 1;
    else return n*b(n-6);
    fi;
  end;
List([0..20], n-> Product([0..n], j-> b(j)) ); # G. C. Greubel, Aug 22 2019

b:=func< n | n le 6 select n else n*Self(n-6) >;
[1] cat [(&*[b(j): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 22 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 22 2019

b[n_]:= b[n]= If[n<1, 1, n*b[n-6]]; a[n_]:= Product[b[j], {j,0,n}];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Aug 22 2019 *)

b(n)=if(n<1, 1, n*b(n-6));
vector(20, n, n--; prod(j=0,n, b(j)) ) \\ G. C. Greubel, Aug 22 2019

def b(n):
    if (n<1): return 1
    else: return n*b(n-6)
[product(b(j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 22 2019

a(n) = Product_{j=0..n} j!!!!!!.
a(n) = Product_{j=0..n} j!6.
a(n) = Product_{j=0..n} A085158(j).
a(n) = n!!!!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2.
a(n) = n*(n-6)!!!!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2.

A117607 Integer complexity of n represented with {1,+,!} and parentheses, where ! can be concatenated for multifactorials.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 4, 4, 5, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 6, 4, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 3, 4, 5, 5

Offset: 1

Jonathan Vos Post, Apr 06 2006

nonn

Using the set of symbols {1, +, !} and parentheses, how many 1's does it take to represent n? "!!" is double factorial, "!!!" is triple factorial and so forth.

lim inf = 3, lim sup = infinity. What is the average behavior of this sequence? - Charles R Greathouse IV, Jun 15 2012

			a(1) = 1 because there is one 1 in "1".
a(2) = 2 because "1 + 1".
a(6) = 3 because "(1+1+1)!".
a(7) = 4 because "(1+1+1)!+1".
a(8) = 4 because "(1+1+1+1)!!" using double factorial.
a(12) = 3 because "((1+1+1)!)!!!!" using quadruple factorial.
a(15) = 5 because "(1+1+1+1+1)!!" using double factorial.
a(16) = 4 because "((1+1+1+1)!!)!!!!!!" using double factorial and sextuple factorial.
a(24) = 3 because "(((1+1+1)!)!!!!)!!!!!!!!!!" using quadruple factorial and decuple factorial.
a(36) = 3 because "(((1+1+1)!)!!!!)!!!!!!!!!" using quadruple factorial and nonuple factorial.

Ed Pegg, Jr., Integer Complexity
Eric Weisstein's World of Mathematics, Multifactorial.
Index to sequences related to the complexity of n

n! = A000142. n!! = A006882. n!!! = A007661. n!!!! = A007662. n!!!!! = A085157. n!!!!!! = A085158. n!!!!!!! = A114799. n!!!!!!!! = A114800. n!!!!!!!!! = A114806.

cp's OEIS Frontend

A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

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A114806 Nonuple factorial, 9-factorial, n!9, n!!!!!!!!!.

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A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.

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A288327 Decuple factorial, 10-factorial, n!10, n!!!!!!!!!!.

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A085149 Numbers k such that k!!!!! - 1 is prime.

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A129116 Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.

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