A055204 -id:A055204 - cp's OEIS frontend

A249017 Records in A055460 (number of primes dividing the squarefree part of n!).

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 85, 86, 87, 88, 91, 93, 95, 98, 99, 100

Offset: 1

Max Alekseyev, Oct 19 2014

nonn

Max Alekseyev, Table of n, a(n) for n = 1..100000

Cf. A055460, A055204, A249016 (indices of records).

np=vector(10^6); b=-1; r=0; for(n=1,10^6, f=factor(n); for(i=1,matsize(f)[1], if(f[i,2]%2, r += (-1)^np[f[i,1]]; np[f[i,1]]=1-np[f[i,1]]; )); if(r>b,b=r;print1(b,", ")) )

A249831 A(n,n) = 1, A(n,k) = A(n,k+1)k / gcd(A(n,k+1),k)^2 if n>k, A(n,k) = A(n,k-1)k / gcd(A(n,k-1),k)^2 if n=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 6, 1, 2, 6, 3, 2, 6, 30, 12, 1, 6, 6, 5, 60, 4, 3, 6, 30, 35, 10, 20, 1, 12, 30, 5, 280, 70, 30, 5, 4, 60, 5, 35, 2520, 140, 210, 30, 1, 20, 10, 35, 70, 252, 1260, 420, 210, 6, 5, 30, 70, 70, 70, 2772, 126, 420, 420, 42, 1, 30, 210, 35, 70, 7

Offset: 1

Alois P. Heinz, Nov 06 2014

			Square array A(n,k) begins:
:   1,  2,  6,   6,  30,  5,  35, 280, 2520,  252, ...
:   1,  1,  3,  12,  60, 10,  70, 140, 1260,  126, ...
:   2,  2,  1,   4,  20, 30, 210, 420,  420,   42, ...
:   6,  6,  3,   1,   5, 30, 210, 420,  420,   42, ...
:   6,  6, 12,   4,   1,  6,  42,  84,   84,  210, ...
:  30, 30, 60,  20,   5,  1,   7,  56,  504, 1260, ...
:   5,  5, 10,  30,  30,  6,   1,   8,   72,  180, ...
:  35, 35, 70, 210, 210, 42,   7,   1,    9,   90, ...
:  70, 70, 35, 105, 420, 84,  56,   8,    1,   10, ...
:  70, 70, 35, 105, 420, 84, 504,  72,    9,    1, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Column k=1 gives A055204(n-1) for n>1.
Row n=1 gives A008339(k+1).
Main diagonal gives: A000012.

A:= proc(n, k) option remember; `if`(k=n, 1,
      (r-> r*k/igcd(r, k)^2)(A(n, k+`if`(n>k, 1, -1))))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);

A[n_, k_] := A[n, k] = If[k == n, 1, Function[{r}, r*k/GCD[r, k]^2][A[n, k+If[n>k, 1, -1]]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 02 2014, translated from Maple *)

A056194 Characteristic cube divisor of n!: a(n) = A056191(n!).

Original entry on oeis.org

1, 1, 1, 8, 8, 1, 1, 8, 8, 1, 1, 27, 27, 216, 1000, 1000, 1000, 125, 125, 1, 9261, 74088, 74088, 343, 343, 2744, 74088, 216, 216, 125, 125, 1000, 35937000, 4492125, 12326391, 12326391, 12326391, 98611128, 8024024008, 125375375125

Offset: 1

Labos Elemer, Aug 02 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..4054

Cf. A000142, A055229, A000188, A008833, A007913, A055231, A055230, A055772, A055071, A055204, A055773 A056552, A056195.

f[p_, e_] := If[OddQ[e] && e > 1, p^3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 06 2020 *)

a(n) = A056191(A000142(n)). - Amiram Eldar, Sep 06 2020

A078389 Number of different values obtained by evaluating all different parenthesizations of 1/2/3/4/.../n.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 60, 116, 192, 384, 544, 1088, 1736, 2576, 3824, 7648, 10352, 20704, 28096, 40256, 62128, 124256, 155488, 227872, 349248, 470352, 622128, 1244256, 1499232, 2998464, 3796224, 5289920, 8048544, 10668096, 12562752, 25125504

Offset: 1

John W. Layman, May 07 2003

a(n) = 2*a(n-1) if n is an odd prime, because (p/q)/n and p/(q/n)=(p/q)*n give exactly two different values for each of the different values p/q from the parenthesizations of 1/.../n-1 and a(n) <= 2*a(n-1) if n is not a prime. - Alois P. Heinz, Nov 23 2008

Let M(n) be the smallest integer among the a(n) values. It seems that, for n >= 4, M(n) = A055204, the squarefree part of n!. - Giovanni Resta, Dec 16 2012

			For n=4, ((1/2)/3)/4 = 1/24, (1/2)/(3/4) = 2/3, (1/(2/3))/4 = 3/8, 1/((2/3)/4) = 6 and 1/(2/(3/4)) = 3/8, giving 4 different values 1/24, 3/8, 2/3 and 6. Thus a(4) = 4.
a(5) = 2*a(4) = 2*4 = 8, because 5 is a prime; the 8 different values are: 1/120, 3/40, 2/15, 5/24, 6/5, 15/8, 10/3, 30. - _Alois P. Heinz_, Nov 23 2008

Index entries for sequences related to parenthesizing

p:= proc(n) option remember; local x;
      if n<1 then {}
    elif n=1 then {1}
    elif n=2 then {1/2}
    else {seq([x/n, x*n][], x=p(n-1))}
      fi
    end:
a:= n-> nops(p(n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 23 2008

p[0] = {}; p[1] = {1}; p[2] = {1/2}; p[n_] := p[n] = Union[ Flatten[ Table[ {x/n, x*n}, {x, p[n - 1]}]]]; a[n_] := Length[p[n]]; A078389 = Table[an = a[n]; Print[an]; an, {n, 1, 30}] (* Jean-François Alcover, Jan 06 2012, after Alois P. Heinz *)

Corrected a(5)-a(10) and extended a(11)-a(31) by Alois P. Heinz, Nov 23 2008

a(32)-a(37) from Alois P. Heinz, Mar 07 2011

A147661 a(n) = squarefree part of n^n.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 21, 1, 23, 1, 1, 1, 3, 1, 29, 1, 31, 1, 33, 1, 35, 1, 37, 1, 39, 1, 41, 1, 43, 1, 5, 1, 47, 1, 1, 1, 51, 1, 53, 1, 55, 1, 57, 1, 59, 1, 61, 1, 7, 1, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 3, 1, 77, 1, 79, 1, 1, 1, 83, 1, 85, 1, 87, 1, 89

Offset: 1

Artur Jasinski, Nov 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000312, A007913, A055204, A147662.

data = Table[Sqrt[n^n], {n, 1, 200}]; sp = data /. Sqrt[] -> 1; sfp = data/sp /. Sqrt[x] -> x

a(n) = core(n^n); \\ Michel Marcus, Nov 01 2022

a(n) = A007913(A000312(n)). - Michel Marcus, Nov 01 2022

A072118 a(n) = (2*n)!/core(n!)^2/(n+1) where core(x) is the squarefree part of x.

Original entry on oeis.org

1, 2, 5, 224, 672, 2737152, 8895744, 474439680, 130660687872, 4513732853760000, 15798064988160000, 894413525483520000, 3194334019584000000, 27610545531676262400000, 8107146431738442547200000, 7569213421444268241715200000, 27753782545295650219622400000

Offset: 1

Benoit Cloitre, Jun 19 2002

Amiram Eldar, Table of n, a(n) for n = 1..251

Cf. A007913, A055204.

f[p_, e_] := p^Mod[e, 2]; a[1] = 1; a[n_] := (2*n)! / ((Times @@ f @@@ FactorInteger[n!])^2 * (n+1)); Array[a, 17] (* Amiram Eldar, May 01 2025 *)

PARI
```
a(n) = (2*n)!/core(n!)^2/(n+1);
```

A240505 Products of primes the squares of which are Fermi-Dirac divisors of n!

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 2, 5, 5, 10, 10, 70, 210, 210, 210, 35, 35, 14, 14, 154, 154, 462, 2310, 30030, 10010, 715, 715, 4290, 4290, 4290, 4290, 36465, 7293, 4862, 4862, 92378, 277134, 277134, 277134, 1939938, 1939938, 88179, 146965, 6760390, 6760390, 20281170

Offset: 1

Vladimir Shevelev, Apr 06 2014

nonn

Or equivalently, products of primes the squares of which are infinitary divisors of n!.

Rémy Sigrist, Table of n, a(n) for n = 1..5000
Rémy Sigrist, PARI program for A240505

Cf. A050376, A055204, A240502, A240504.

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k,2]); bde = binary(valuation(d, f[k,1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)););); return (1);}
a(n) = {f = factor(n!); for (k=1, #f~, if ((f[k,2] > 1) && isidiv(f[k,1]^2, f), f[k,2]=1, f[k,2]=0);); factorback(f);} \\ Michel Marcus, Feb 15 2016

a(23)-a(32) from Michel Marcus, Feb 15 2016

a(1) = 1 and more terms from Rémy Sigrist, Feb 13 2019

A248779 Rectangular array, by antidiagonals: T(m,n) = greatest (m+1)-th-power-free divisor of n!.

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 6, 6, 2, 1, 30, 3, 6, 2, 1, 5, 15, 24, 6, 2, 1, 35, 90, 120, 24, 6, 2, 1, 70, 630, 45, 120, 24, 6, 2, 1, 70, 630, 315, 720, 120, 24, 6, 2, 1, 7, 210, 2520, 5040, 720, 120, 24, 6, 2, 1, 77, 2100, 280, 1260, 5040, 720, 120, 24, 6, 2, 1, 231

Offset: 1

Clark Kimberling, Oct 14 2014

Row 1:  A055204, greatest squarefree divisor of n!
Row 2:  A145642, greatest cubefree divisor of n!
Row 3:  A248766, greatest 4th-power-free divisor of n!
Rows 4 to 7:  A248769, A248772, A248775, A248778.
(The divisors are here called "greatest" rather than "largest" because the name refers to ">", called "greater than".)

			Northwest corner:
1   2   6   6   30   5    35    70
1   2   6   3   15   90   630   630
1   2   6   24  120  45   315   2520
1   2   6   24  120  720  5040  1260

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A055204, A145642, A248766, A248769, A248772, A248775, A248778, A000142.

f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}]
t = Table[n!/p[m, n], {m, 2, 16}, {n, 1, 16}]; TableForm[t]  (* A248779 array *)
f = Table[t[[n - k + 1, k]], {n, 12}, {k, n, 1, -1}] // Flatten (* A248779 seq. *)

A299700 Squarefree part of 1!2!3!...n!: The product of factorials one through n divided by its largest square divisor.

Original entry on oeis.org

1, 2, 3, 2, 15, 3, 105, 6, 105, 15, 1155, 5, 15015, 70, 1001, 70, 17017, 35, 323323, 7, 138567, 154, 3187041, 231, 3187041, 6006, 1062347, 858, 30808063, 715, 955049953, 1430, 260468169, 12155, 9116385915, 12155, 337306278855, 461890, 8648878945, 46189, 354604036745, 1939938, 15247973580035, 176358

Offset: 1

Graeme McRae, Feb 17 2018

nonn

Smallest number such that a(n)*1!*2!*3!*...*n! is a square.
If n is even, a(2n) = A055204(n).
If n is odd and evil (A129771) then a(2n) = A055204(n)/2.
If n is odd and odious (A092246) then a(2n) = 2*A055204(n).

			1!*2!*3!*4!*5! = 2^8 * 3^3 * 5^1 so a(5) = 3*5 = 15.

Amiram Eldar, Table of n, a(n) for n = 1..2970
Graeme McRae, Evil and Odious Numbers, Factorial, and Superfactorial, Maths Blog, Quora.

Cf. A000178, A007913, A055204, A092246, A129771.

Nest[Append[#, {#, Sqrt[#] /. (c_: 1) a_^(b_: 0) :> (c a^b)^2} &[#[[-1, 1]]*Length[# + 1]!]] &, {{1, 1}}, 44][[All, -1]] (* Michael De Vlieger, Feb 17 2018, after Bill Gosper at A007913 *)
f[n_] := Block[{m = BarnesG[n +2], p = 2}, While[p < n, While[ Mod[m, p^2] == 0, m/=p^2]; p = NextPrime@ p]; m]; Array[f, 42] (* Robert G. Wilson v, Feb 18 2018 *)

a(n) = core(prod(k=1, n, k!)); \\ Michel Marcus, Feb 17 2018

a(n) = A007913(A000178(n)). - Michel Marcus, Feb 17 2018

cp's OEIS Frontend

A249017 Records in A055460 (number of primes dividing the squarefree part of n!).

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A249831 A(n,n) = 1, A(n,k) = A(n,k+1)*k / gcd(A(n,k+1),k)^2 if n>k, A(n,k) = A(n,k-1)*k / gcd(A(n,k-1),k)^2 if n=1, k>=1, read by antidiagonals.

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A056194 Characteristic cube divisor of n!: a(n) = A056191(n!).

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A078389 Number of different values obtained by evaluating all different parenthesizations of 1/2/3/4/.../n.

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Maple

Mathematica

Extensions

A147661 a(n) = squarefree part of n^n.

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A072118 a(n) = (2*n)!/core(n!)^2/(n+1) where core(x) is the squarefree part of x.

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A240505 Products of primes the squares of which are Fermi-Dirac divisors of n!

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Author

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Programs

PARI

Extensions

A248779 Rectangular array, by antidiagonals: T(m,n) = greatest (m+1)-th-power-free divisor of n!.

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Keywords

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Examples

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Programs

Mathematica

A299700 Squarefree part of 1!*2!*3!*...*n!: The product of factorials one through n divided by its largest square divisor.

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A249831 A(n,n) = 1, A(n,k) = A(n,k+1)k / gcd(A(n,k+1),k)^2 if n>k, A(n,k) = A(n,k-1)k / gcd(A(n,k-1),k)^2 if n=1, k>=1, read by antidiagonals.

A299700 Squarefree part of 1!2!3!...n!: The product of factorials one through n divided by its largest square divisor.