A288327 -id:A288327 - cp's OEIS frontend

A288369 Primes of the form k!10 + 1, where k!10 is the decuple factorial number (A288327).

2, 3, 5, 7, 11, 97, 45697, 4133377, 4060162871525377, 294880677941535796547813377, 46001385758879584261458886657, 7636230035974010987402175184897, 35568742809600000000000000000001, 18067746635539299564851337380417765377

Offset: 1

Robert Price, Jun 08 2017

nonn

Robert Price, Table of n, a(n) for n = 1..21
Joe McLean, Interesting Sources of Probable Primes.

Cf. A204656, A288327.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 10] + 1, {i, 0, 100}], PrimeQ[#]&]

a(n) = 1 + A288327(A204656(n+1)). - Elmo R. Oliveira, Feb 26 2025

A288370 Primes of the form k!10 + 2, where k!10 is the decuple factorial number (A288327).

Original entry on oeis.org

3, 5, 7, 11, 13, 41, 173, 233, 1877, 293603, 318482201, 3047775608243, 22045250515087152640289, 1302844523174285888671877, 930620100318118916029523201, 4831436058626442432403564453127, 2060356301148292483532951454058361, 9936127455089061347552058319626135203

Offset: 1

Robert Price, Jun 08 2017

nonn

Robert Price, Table of n, a(n) for n = 1..29
Henri and Renaud Lifchitz, PRP Records.Search for n!10+2.
Joe McLean, Interesting Sources of Probable Primes.
OpenPFGW Project, Primality Tester.

Cf. A204657, A288327.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 10] + 2, {i, 0, 100}], PrimeQ[#]&]
Select[Table[Times@@Range[n,1,-10]+2,{n,200}],PrimeQ] (* Harvey P. Dale, May 26 2025 *)

a(n) = 2 + A288327(A204657(n+1)). - Elmo R. Oliveira, Feb 26 2025

A289754 Primes of the form k!10 - 1, where k!10 is the decuple factorial number (A288327).

Original entry on oeis.org

2, 3, 5, 7, 23, 199, 239999, 7354367, 719999999, 2287853567, 50399999999, 43193222037503, 199094626418687, 17254743876205585224626400657407, 620448401733239439359999999999999999999999999999, 17999257015554062631539656405422901164857242675052543

Offset: 1

Robert Price, Jul 11 2017

nonn

Robert Price, Table of n, a(n) for n = 1..18
Henri and Renaud Lifchitz, PRP Records.Search for n!10-1.
Joe McLean, Interesting Sources of Probable Primes.
OpenPFGW Project, Primality Tester.

Cf. A204658, A288327.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 10] - 1, {i, 2, 100}], PrimeQ[#]&]

a(n) = A288327(A204658(n)) - 1. - Elmo R. Oliveira, Apr 14 2025

A289861 Primes of the form k!10 - 2, where k!10 is the decuple factorial number (A288327).

Original entry on oeis.org

2, 3, 5, 7, 37, 73, 229, 1873, 4957, 7159, 29599, 293599, 2953123, 9476647, 1643049664639, 30501459767616059381067752838134765623, 75901465778001946800870398616055887199, 168354282831355577455162448188672669399, 73004868196707960876168803928840928829809

Offset: 1

Robert Price, Jul 13 2017

nonn

Robert Price, Table of n, a(n) for n = 1..32
Henri and Renaud Lifchitz, PRP Records.Search for n!10-2.
Joe McLean, Interesting Sources of Probable Primes.
OpenPFGW Project, Primality Tester.

Cf. A283559, A288327.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 10] - 2, {i, 3, 100}], PrimeQ[#]&]
Select[Table[Times@@Range[n,1,-10]-2,{n,300}],PrimeQ] (* Harvey P. Dale, Sep 09 2021 *)

a(n) = A288327(A283559(n)) - 2. - Elmo R. Oliveira, Feb 25 2025

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

cp's OEIS Frontend

A288369 Primes of the form k!10 + 1, where k!10 is the decuple factorial number (A288327).

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A288370 Primes of the form k!10 + 2, where k!10 is the decuple factorial number (A288327).

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Keywords

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Crossrefs

Programs

Mathematica

Formula

A289754 Primes of the form k!10 - 1, where k!10 is the decuple factorial number (A288327).

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Mathematica

Formula

A289861 Primes of the form k!10 - 2, where k!10 is the decuple factorial number (A288327).

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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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