A007662 -id:A007662 - cp's OEIS frontend

A084980 Triangle of (multi)factorials: n-th row is (n+1)!!... {n "!"s}, (n+1)!... {n-1 "!"s}, ..., (n+1)!.

2, 3, 6, 4, 8, 24, 5, 10, 15, 120, 6, 12, 18, 48, 720, 7, 14, 21, 28, 105, 5040, 8, 16, 24, 32, 80, 384, 40320, 9, 18, 27, 36, 45, 162, 945, 362880, 10, 20, 30, 40, 50, 120, 280, 3840, 3628800, 11, 22, 33, 44, 55, 66, 231, 880, 10395, 39916800, 12, 24, 36, 48, 60, 72, 168

Offset: 1

Rick L. Shepherd, Jul 16 2003

			Triangle begins
Row 1: 2! = 2
Row 2: 3!! = 3*1 = 3, 3! = 3*2*1 = 6
Row 3: 4!!! = 4*1 = 4, 4!! = 4*2 = 8, 4! = 4*3*2*1 = 24,
Row 4: 5!!!! = 5*1 = 5, 5!!! = 5*2 = 10, 5!! = 5*3 = 15, 5! = 5*4*3*2*1 = 120,
...

Ivan Neretin, Table of n, a(n) for n = 1..4950
Eric Weisstein's World of Mathematics, Factorial.
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142 (n!), A006882 (n!!), A007661 (n!!!), A007662 (n!!!!).

Flatten[Table[Times @@ Range[n, 1, -i], {n, 12}, {i, n - 1, 1, -1}]] (* Ivan Neretin, May 09 2015 *)

A085147 Numbers k such that k!!!! - 1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 12, 16, 22, 24, 54, 56, 98, 152, 156, 176, 256, 454, 460, 720, 750, 770, 800, 1442, 2846, 5920, 7066, 12778, 19978, 22726, 25938, 27780, 36072, 39746, 48244

Offset: 1

Hugo Pfoertner, Jun 25 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).

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!4-1.

Cf. A007662 (quadruple factorials), A085146.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[MultiFactorial[#, 4] - 1] & ] (* Robert Price, Apr 19 2019 *)
Select[Range[48300],PrimeQ[Times@@Range[#,1,-4]-1]&] (* Harvey P. Dale, Aug 11 2020 *)

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

A114790 Cumulative product of quintuple factorial A085157.

Original entry on oeis.org

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.

A173574 4-Factorions: equal to the sum of the quadruple factorials of their digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 49

Offset: 1

Paolo P. Lava, Feb 22 2010

			49 -> 4!!!! + 9!!!! = 4 + 9*5 = 4 +45 = 49.

Cf. A007662, A014080, A097653, A173573, A173575-A173577

P:=proc(n,m) local a,b,i,j,k,x,w; for i from 1 by 1 to n do a:=0; b:=0; w:=0; k:=i; while k>0 do w:=k-(trunc(k/10)*10); j:=w; x:=w-m; if w=0 then b:=1; else while x>0 do j:=j*x; x:=x-m; od; b:=j; fi; a:=a+b; k:=trunc(k/10); od; if a=i then lprint(i,a); fi; od; end: P(1000,4);

qfd[n_]:=Times@@Range[n,1,-4]; Select[Range[50],Total[qfd/@ IntegerDigits[ #]] == #&] (* Harvey P. Dale, Dec 15 2018 *)

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.

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).

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.

A153189 Triangle T(n,k) = Product_{j=0..k} n*j+1.

Original entry on oeis.org

1, 1, 2, 1, 3, 15, 1, 4, 28, 280, 1, 5, 45, 585, 9945, 1, 6, 66, 1056, 22176, 576576, 1, 7, 91, 1729, 43225, 1339975, 49579075, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625

Offset: 0

Roger L. Bagula, Dec 20 2008

Row sums are: {1, 3, 19, 313, 10581, 599881, 50964103, 6047094369, 954249517513, 193146844030201, 48762935887310811,...}. [Corrected by M. F. Hasler, Oct 28 2014]
This is the lower left triangle of the array A142589. - M. F. Hasler, Oct 28 2014
Row n is a subset of the n-fold factorial sequence for k=0..n. For example, T(8,0..8) is A045755(1..9).  These sequences are listed for n=0..10 in A256268. - Georg Fischer, Feb 15 2020

			Triangle begins as:
  1;
  1, 2;
  1, 3,  15;
  1, 4,  28,  280;
  1, 5,  45,  585,   9945;
  1, 6,  66, 1056,  22176,  576576;
  1, 7,  91, 1729,  43225, 1339975,  49579075;
  1, 8, 120, 2640,  76560, 2756160, 118514880,  5925744000;
  1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000165, A006882, A007661, A007662, A008544.

Cf. A000142 (row 2), A001147 (3), A007559 (4), A007696 (5), A008548 (6), A008542 (7), A045754 (8), A045755 (9), A045756 (10), A144773 (11), A256268 (combined table).

[(&*[n*j+1: j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 15 2020

seq(seq(mul(n*j+1, j=0..k), k=0..n), n=0..10); # G. C. Greubel, Feb 15 2020

T[n_, k_]= If[n==0 && k==0, 1, Product[n*j+1, {j,0,k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2020 *)
T[n_, k_]:= T[n, k]= If[k<2, 1+k*n, ((1+n*k)*T[n, k-1] + (1+n*k)*(1+n*(k-1))* T[n, k-2])/2]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Georg Fischer, Feb 17 2020 *)

T(n,k)=prod(j=0,k,n*j+1) \\ M. F. Hasler, Oct 28 2014

[[ product(n*j+1 for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 15 2020

T(n, k) = n^(k+1)*Pochhammer(1/n, k+1).
From Vaclav Kotesovec, Oct 10 2016: (Start)
For fixed n > 0:
T(n, k) ~ sqrt(2*Pi) * n^k * k^(k + 1/2 + 1/n) / (Gamma(1 + 1/n) * exp(k)).
T(n, k) ~ k! * n^k * k^(1/n) / Gamma(1 + 1/n).
(End)
T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*Stirling1(k+1,j)*n^(k-j+1). - G. C. Greubel, Feb 17 2020
T(n, k) = ((1+n*k)*T(n, k-1) + (1+n*k)*(1+n*(k-1))*T(n, k-2))/2. - Georg Fischer, Feb 17 2020

Edited and row 0 added by M. F. Hasler, Oct 28 2014

A337354 a(n) is the numerator of Product_{i=0..n-1} (n-i)^((-1)^ceiling(i/2)).

Original entry on oeis.org

1, 2, 3, 2, 5, 9, 7, 40, 45, 7, 308, 48, 975, 539, 88, 1664, 1105, 24255, 13376, 56576, 41769, 48279, 55936, 226304, 348075, 370139, 671232, 870400, 2082925, 4283037, 13872128, 80773120, 343682625, 4023459, 1553678336, 1900544, 14411758075, 59457783, 1471905792, 1406402560

Offset: 1

Devansh Singh, Aug 24 2020

a(n) is the numerator of (n/(n-1)) * ((n-3)/(n-2)) * ((n-4)/(n-5)) ...

			a(n)/A337355(n) equals 1, 2, 3/2, 2/3, 5/6, 9/5, 7/5, 40/63, 45/56, 7/4 ...
a(4) = numerator of (4*1)/(3*2) = numerator of 2/3 = 2.
a(5) = numerator of (5*2)/(4*3) = numerator of 5/6 = 5.
                      12  *   9*8  *  5*4  *  1
a(12) = numerator of --------------------------- = 48.
                        11*10  *  7*6  *  3*2

Cf. A337355 (denominators).

Cf. A076390, A085565, A007662.

a(n) = {numerator(prod(i=0, n-1, (n-i)^(-1)^((i+1)\2)))} \\ Andrew Howroyd, Aug 24 2020

a(n) = numerator of (n*A337355(n-2))/(a(n-2)*(n-1)) for n>=3.

Conjecture: a(4*n)/A337355(4*n) ~ 0.5990701173677... (=A076390). - Andrew Howroyd, Aug 25 2020

Terms a(31) and beyond from Andrew Howroyd, Aug 25 2020

cp's OEIS Frontend

A084980 Triangle of (multi)factorials: n-th row is (n+1)!!... {n "!"s}, (n+1)!... {n-1 "!"s}, ..., (n+1)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A085147 Numbers k such that k!!!! - 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A114790 Cumulative product of quintuple factorial A085157.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A173574 4-Factorions: equal to the sum of the quadruple factorials of their digits in base 10.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A114423 Multifactorial array read by ascending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A114796 Cumulative product of sextuple factorial A085158.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs