A114800 -id:A114800 - cp's OEIS frontend

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

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

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.

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

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A114423 Multifactorial array read by ascending antidiagonals.

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

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Crossrefs