A000178 -id:A000178 - cp's OEIS frontend

A203432 a(n) = A203430(n)/A000178(n) where A000178=(superfactorials).

1, 2, 3, 15, 45, 540, 3402, 96228, 1299078, 85739148, 2507870079, 383704122087, 24487299427734, 8645900336407620, 1209640056157393380, 982320774834892454820, 302358334494179897593596, 563293577162657149216869348

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..100
R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Cf. A000178, A032766, A203430, A203431.

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
A203432:= func< n | n eq 1 select 1 else (&*[(&*[k-j+Floor((k+1)/2)-Floor((j+1)/2): j in [0..k-1]]) : k in [1..n-1]])/Barnes(n) >;
[A203432(n): n in [1..25]]; // G. C. Greubel, Sep 20 2023

f[j_]:= j + Floor[j/2]; z = 20;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]             (* A203430 *)
Table[v[n+1]/v[n], {n,z}]      (* A203431 *)
Table[v[n]/d[n], {n,z}]        (* this sequence *)

def barnes(n): return product(factorial(j) for j in range(n))
def A203432(n): return product(product(k-j+((k+1)//2)-((j+1)//2) for j in range(k)) for k in range(1,n))/barnes(n)
[A203432(n) for n in range(1,31)] # G. C. Greubel, Sep 20 2023

A336497 Numbers that cannot be written as a product of superfactorials A000178.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

First differs from A336426 in having 360.

			The sequence of terms together with their prime indices begins:
     3: {2}        22: {1,5}        39: {2,6}
     5: {3}        23: {9}          40: {1,1,1,3}
     6: {1,2}      25: {3,3}        41: {13}
     7: {4}        26: {1,6}        42: {1,2,4}
     9: {2,2}      27: {2,2,2}      43: {14}
    10: {1,3}      28: {1,1,4}      44: {1,1,5}
    11: {5}        29: {10}         45: {2,2,3}
    13: {6}        30: {1,2,3}      46: {1,9}
    14: {1,4}      31: {11}         47: {15}
    15: {2,3}      33: {2,5}        49: {4,4}
    17: {7}        34: {1,7}        50: {1,3,3}
    18: {1,2,2}    35: {3,4}        51: {2,7}
    19: {8}        36: {1,1,2,2}    52: {1,1,6}
    20: {1,1,3}    37: {12}         53: {16}
    21: {2,4}      38: {1,8}        54: {1,2,2,2}

A093373 is the version for factorials, with complement A001013.
A336426 is the version for superprimorials, with complement A181818.
A336496 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178(n).
A174605 is the maximum prime multiplicity in A000178(n).
A303279 counts prime factors (with multiplicity) of superprimorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[100],facsusing[Rest[Array[supfac,30]],#]=={}&]

A349081 Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!2!...*k! is the superfactorial of k.

Original entry on oeis.org

8, 14, 16, 32, 48, 72, 96, 128, 160, 200, 240, 288, 336, 392, 448, 512, 574, 576, 648, 720, 800, 880, 968, 1056, 1152, 1248, 1352, 1456, 1568, 1680, 1800, 1920, 2048, 2176, 2312, 2448, 2592, 2736, 2888, 3040, 3200, 3360, 3528, 3696, 3872, 4048, 4232, 4416, 4608, 4800, 5000

Offset: 1

Bernard Schott, Dec 01 2021

nonn

This sequence is the union of three infinite and disjoint subsequences:
-> Numbers k = 8t^2 > 0 (A139098); for these numbers, m_1 = k/2 - 1 = 4t^2-1 < m_2 = k/2 = 4t^2  (see example for k = 8).
-> Numbers k = 8t*(t+1) (A035008); for these numbers, m_1 = k/2 = 4t(t+1) < m_2 =  k/2 + 1 = (2t+1)^2 (see example for k = 16).
-> Even numbers of the form 2t^2-4, t>1 in A001541 (A349766); for these numbers, m_1 = k/2 + 1 = t^2 - 1 <  m_2 = k/2 + 2 = t^2 (see example for k = 14).
See A348692 for further information.

			For k = 8, 8$ / 2! is not a square, but m_1 = 3 because 8$ / 3! = 29030400^2 and m_2 = 4 because 8$ / 4! = 14515200^2.
For k = 14, m_1 = 8 because 14$ / 8! = 1309248519599593818685440000000^2 and m_2 = 9 because 14$ / 9! = 436416173199864606228480000000^2.
For k = 16, m_1 = 8 because 16$ / 8! = 6848282921689337839624757371207680000000000^2 and m_2 = 9 because 16$ / 9! = 2282760973896445946541585790402560000000000^2.

Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, pages 346-352.

Cf. A001541, A035008, A139098, A348692, A349080.

Subsequence of A349079.

Do[j=0;l=1;g=BarnesG[k+2];While[j<2&&l<=k,If[IntegerQ@Sqrt[g/l!],j++];l++];If[j==2,Print@k],{k,5000}] (* Giorgos Kalogeropoulos, Dec 02 2021 *)

sf(n) = prod(k=2, n, k!); \\ A000178
isok(m) = if (!(m%2), my(s=sf(m)); #select(issquare, vector(4, k, s/(m/2+k-2)!), 1) == 2); \\ Michel Marcus, Dec 04 2021

A203434 a(n) = A203433(n)/A000178(n) where A000178=(superfactorials).

Original entry on oeis.org

1, 1, 3, 6, 45, 189, 3402, 30618, 1299078, 25332021, 2507870079, 106698472452, 24487299427734, 2283997201168644, 1209640056157393380, 248218139523497121576, 302358334494179897593596, 136861610819571430116630660

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..100
R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Cf. A000178, A007494, A014402, A203432, A203433.

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
f:= func< k | (&*[k+1-j+Floor((k+2)/2)-Floor((j+1)/2): j in [1..k]]) >;
[1] cat [(&*[f(k): k in [1..n-1]])/Barnes(n): n in [2..20]]; // G. C. Greubel, Sep 19 2023

f[j_]:= j + Floor[(j+1)/2]; z = 20;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]             (* A203433 *)
Table[v[n+1]/v[n], {n,z}]      (* A014402 *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)

def barnes(n): return product(factorial(j) for j in range(n))
def f(k): return product(k-j+(k//2)-(j//2) for j in range(k))
[product(f(k) for k in range(1, n) )//barnes(n) for n in range(1,31)] # G. C. Greubel, Sep 19 2023

A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

Original entry on oeis.org

1, 3, 30, 1050, 132300, 61122600, 104886381600, 674943865596000, 16407885372638760000, 1515727634953623371280000, 534621388490302221024396480000, 722849817707190846398223943885440000, 3759035907022704558524683975387453632000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A000178, A006963, A076756, A093883, A296590.

[(&*[Binomial(2*n-k,k): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023

(* First program *)
f[j_]:= j; z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]           (* A093883 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A006963 *)
Table[v[n]/d[n], {n,20}]     (* A203469 *)
(* Second program *)
Table[Product[Binomial[2*n-j,j], {j,n}], {n,20}] (* G. C. Greubel, Aug 29 2023 *)

[product(binomial(2*n-j,j) for j in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023

a(n) = Product_{i=1..n} binomial(2n-i,i).  - Enrique Pérez Herrero, Feb 20 2013
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = (2^n/sqrt(Pi))^n*BarnesG(n+3/2)/(BarnesG(n+2)*BarnesG(3/2)).
a(n) = (n!/2^(n-1))*Product_{j=1..n-1} Catalan(j). (End)
a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^(n/2 + 1/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 26 2023

A203474 a(n) = A203472(n) / A000178(n-1), where A000178 are the superfactorials.

Original entry on oeis.org

1, 7, 252, 41580, 29729700, 89278289100, 1104908105901600, 55674109640169820800, 11329124570678156834592000, 9258047307912482983660236480000, 30262334718212007877669234596364800000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A000178, A093883, A203472, A203473.

[(&*[ Binomial(2*j+3, j+4): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 27 2023

(* First program *)
f[j_]:= j+2; z=16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,n}]  (* A000178(n-1) *)
Table[v[n], {n,z}]              (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]     (* A203473 *)
Table[v[n]/d[n], {n,20}]        (* A203474 *)
(* Second program *)
Table[Product[Binomial[2*j+3, j+4], {j,n}], {n,20}] (* G. C. Greubel, Aug 27 2023 *)

[product( binomial(2*j+5,j+5) for j in range(n) ) for n in range(1,20)] # G. C. Greubel, Aug 27 2023

a(n) ~ 3*A^(3/2) * 2^(n^2 + 4*n + 185/24) * exp(n/2 - 1/8) / (Pi^(n/2 + 3/2) * n^(n/2 + 59/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 09 2021
From G. C. Greubel, Aug 27 2023: (Start)
a(n) = Product_{j=1..n} binomial(2*j+3, j+4).
a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/( BarnesG(n +1)*BarnesG(n+6)*BarnesG(7/2)). (End)

Definition corrected by Vaclav Kotesovec, Apr 09 2021

A203510 a(n) = A203482(n) / A000178(n).

Original entry on oeis.org

1, 3, 84, 273000, 3046699656000, 5996663814749677445376000, 160771799453017261771769947549079938007040000, 6351968589735888467306807912855132014808202373395298410963148996608000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term of the sequence is an integer.

G. C. Greubel, Table of n, a(n) for n = 1..16

Cf. A000142, A000178, A074962, A093883, A203482, A203483.

BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203510:= func< n | n eq 1 select 1 else (&*[(&*[Factorial(j) + Factorial(k): k in [1..j-1]]): j in [2..n]])/BarnesG(n+1) >;
[A203510(n): n in [1..13]]; // G. C. Greubel, Feb 24 2024

f[j_] := j!; z = 10;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
Table[v[n], {n, 1, z}]                 (* A203482 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203483 *)
Table[v[n]/d[n], {n, 1, 10}]           (* this sequence *)
Table[Product[j! + k!, {j, 1, n}, {k, 1, j-1}] / BarnesG[n+1], {n, 1, 10}] (* Vaclav Kotesovec, Nov 20 2023 *)

def BarnesG(n): return product(factorial(j) for j in range(1,n-1))
def A203510(n): return product(product(factorial(j)+factorial(k) for k in range(1,j)) for j in range(1,n+1))/BarnesG(n+1)
[A203510(n) for n in range(1,14)] # G. C. Greubel, Feb 24 2024

a(n) ~ c * A * n^(n^3/3 - n^2/4 - 7*n/12 + 17/24) * (2*Pi)^(n^2/4 - 3*n/4) / exp(4*n^3/9 - 7*n^2/8 - n + 1/12), where A is the Glaisher-Kinkelin constant A074962 and c = 0.488888619502150098591650327163991582267254151817880403495924251381414248582... (from A203482). - Vaclav Kotesovec, Nov 20 2023

A203520 v(n)/A000178(n); v=A203518 and A000178=(superfactorials).

Original entry on oeis.org

1, 3, 30, 1680, 900900, 9535125600, 4122929827336320, 161481256755920962660800, 1289130207153926967849156327590400, 4850265693548396005370498087328884780717568000, 20141097979706537636828034511787661382412368790843921121216000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term of A203520 is an integer.

Cf. A203518, A000045, A000178, A093883.

f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
Table[v[n], {n, 1, z}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

A203523 v(n)/A000178(n); v=A203521 and A000178=(superfactorials).

Original entry on oeis.org

1, 5, 140, 25200, 55036800, 951035904000, 222618484408320000, 440079343769868042240000, 12254449406615745504215040000000, 7909254579604123100510930935480320000000, 48073937540175558516708030362614204937011200000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term of A203523 is an integer.

Cf. A203522, A203523, A000040, A093883.

f[j_] := Prime[j]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
Table[v[n], {n, 1, z}]                (* A203521 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203522 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203523 *)

A203526 v(n)/A000178(n); v=A203524 and A000178=(superfactorials).

Original entry on oeis.org

1, 8, 480, 322560, 1857945600, 137339338752000, 90498933234597888000, 806410654352196092559360000, 151104996166246050391298219704320000, 278316545034703677313682486677538340864000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term of A203526 is an integer.

Cf. A203524, A203525, A093883, A203317.

f[j_] := Prime[j + 1]; z = 17;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]    (* A000178 *)
Table[v[n], {n, 1, z}]                   (* A203524 *)
Table[v[n + 1]/(8 v[n]), {n, 1, z - 1}]  (* A203525 *)
Table[v[n]/d[n], {n, 1, 20}]             (* A203526 *)

cp's OEIS Frontend

A203432 a(n) = A203430(n)/A000178(n) where A000178=(superfactorials).

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A336497 Numbers that cannot be written as a product of superfactorials A000178.

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A349081 Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.

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A203434 a(n) = A203433(n)/A000178(n) where A000178=(superfactorials).

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A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

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A203474 a(n) = A203472(n) / A000178(n-1), where A000178 are the superfactorials.

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A203510 a(n) = A203482(n) / A000178(n).

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A203520 v(n)/A000178(n); v=A203518 and A000178=(superfactorials).

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A349081 Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!2!...*k! is the superfactorial of k.