A055772 -id:A055772 - cp's OEIS frontend

A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390, 19835652870, 2333606220

Offset: 0

Labos Elemer, Jul 25 2000

nonn

a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm(A001405(n-1), A001405(n)) (for n>0).
A055773(n) divides a(n), A001316(floor(n/2)) divides a(n).
Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012
a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]

			a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Didier Guillet, On swinging factorials and the lonely runner conjecture (Text in French).
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Orbitals.
Peter Luschny, Swinging Factorial.

Bisections are A000984 and A002457.

Cf. A000142, A001405, A000188, A055772, A056042, A211226, A162246, A189231, A212303.

[(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011

SeriesCoeff := proc(s,n) series(s(w,n),w,n+2);
convert(%,polynom); coeff(%,w,n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)),k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0,1,n^irem(n,2)*(4/n)^irem(n+1,2)*a2(n-1)) end;
a3 := n -> n!/iquo(n,2)!^2;
g4 := z -> BesselI(0,2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4,n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5,n);
g6 := (z,n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6,n);
a7 := n -> combinat[multinomial](n,floor(n/2),n mod 2,floor(n/2));
h := n -> binomial(n,floor(n/2)); # A001405
a8 := n -> ilcm(h(n-1),h(n));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 02 2010 *)
f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 10 2010 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)

a(n)=n!/(n\2)!^2 \\ Charles R Greathouse IV, May 02 2011

def A056040():
    r, n = 1, 0
    while True:
        yield r
        n += 1
        r *= 4/n if is_even(n) else n
a = A056040(); [next(a) for i in range(36)]  # Peter Luschny, Oct 24 2013

a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
E.g.f.: (1+x)*BesselI(0, 2*x). - Vladeta Jovovic, Jan 19 2004
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
a(2n+1) = A046212(2n+1) = A100071(2n+1). - M. F. Hasler, Jan 25 2012
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012
D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012
Sum_{n>=0} 1/a(n) = 4/3 + 8*Pi/(9*sqrt(3)). - Alexander R. Povolotsky, Aug 18 2012
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013
a(n) = Sum_{k=0..n} A189231(n, 2*k). (Cf. A212303 for the odd case.) - Peter Luschny, Oct 30 2013
a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014
a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015
Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022

Extended and edited by Peter Luschny, Jun 28 2009

A055204 Squarefree part of n!: n! divided by its largest square divisor.

Original entry on oeis.org

1, 2, 6, 6, 30, 5, 35, 70, 70, 7, 77, 231, 3003, 858, 1430, 1430, 24310, 12155, 230945, 46189, 969969, 176358, 4056234, 676039, 676039, 104006, 312018, 44574, 1292646, 1077205, 33393355, 66786710, 2203961430, 64822395, 90751353, 90751353

Offset: 1

Labos Elemer, Jun 19 2000

nonn

Smallest number such that n!*a(n) is a square.

			10! = 518400*7 = 7*(720)^2, so a(10) = 7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..3386 (next term has 1001 digits)
Kevin A. Broughan, Asymptotic Order of the Square-free Part of N!, Integers, 2 (2002), Article A.10.
Rafael Jakimczuk, On the h-th free part of the factorial, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.

Cf. A000142, A007913, A055071, A249831.

f[p_, e_] := p^Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 01 2024 *)
a[n_] := Block[{fi = Transpose@ FactorInteger[n!]}, Times @@ (fi[[1]]^Mod[fi[[2]], 2])]; Array[a, 40] (* Robert G. Wilson v, Nov 17 2024 *)

a(n)=core(n!) \\ Charles R Greathouse IV, Apr 03 2012

a(n) = A007913(n!) = n!/A055071(n) = A000142(n)/A055071(n).
log a(n) ~ n log 2. - Charles R Greathouse IV, Apr 03 2012
sqrt(n!) = A055772(n) * sqrt(a(n)). - Alonso del Arte, Feb 16 2015

A055071 Largest square dividing n!.

Original entry on oeis.org

1, 1, 1, 4, 4, 144, 144, 576, 5184, 518400, 518400, 2073600, 2073600, 101606400, 914457600, 14631321600, 14631321600, 526727577600, 526727577600, 52672757760000, 52672757760000, 6373403688960000, 6373403688960000, 917770131210240000, 22944253280256000000

Offset: 1

Labos Elemer, Jun 13 2000

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Cf. A000188, A008833, A055772.

seq(expand(numtheory[nthpow](n!, 2)), n=1..26); # Peter Luschny, Apr 03 2013

a[n_] := Select[Reverse @ Divisors[n!], IntegerQ[Sqrt[#]] &, 1] // First; a /@ Range[23] (* Jean-François Alcover, May 19 2011 *)
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n!]); Array[a, 30] (* Amiram Eldar, Jul 26 2024 *)

a(n)=core(n!,2)[2]^2 \\ Charles R Greathouse IV, Apr 04 2012

from math import prod
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A055071_gen(): # generator of terms
    c = Counter()
    for i in count(1):
        c += Counter(factorint(i))
        yield prod(p**(e-(e&1)) for p, e in c.items())
A055071_list = list(islice(A055071_gen(),30)) # Chai Wah Wu, Jul 27 2024

a(n) = A008833(n!).
log a(n) ~ n log n. - Charles R Greathouse IV, Apr 04 2012
a(n) = A055772(n)^2. - Amiram Eldar, Jul 26 2024

More terms from James Sellers, Jun 20 2000

A055993 Number of square divisors of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 6, 8, 12, 30, 30, 36, 36, 72, 96, 128, 128, 180, 180, 300, 300, 600, 600, 864, 1152, 2304, 2688, 4368, 4368, 5376, 5376, 6144, 6144, 13056, 16320, 19440, 19440, 38880, 43200, 48000, 48000, 64000, 64000, 100800, 133056, 278784, 278784

Offset: 1

Labos Elemer, Jul 21 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000142, A000005, A046951, A055772, A000188.

Table[Count[Divisors[n!], d_ /; IntegerQ@ Sqrt@ d], {n, 30}] (* Michael De Vlieger, Feb 05 2017 *)

a(n) = sumdiv(n!, d, issquare(d)); \\ Michel Marcus, Dec 26 2013

a(n) = A046951(n!) = A000005(A055772(n)) = A000005(A000188(A000142(n))).

A056038 Largest factorial k! such that (k!)^2 divides n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 24, 24, 720, 720, 720, 720, 5040, 5040, 40320, 40320, 362880, 362880, 3628800, 3628800, 39916800, 39916800, 479001600, 479001600, 6227020800, 6227020800, 1307674368000, 1307674368000, 1307674368000, 1307674368000, 20922789888000

Offset: 0

Labos Elemer, Jul 25 2000

nonn

This is neither floor(n/2)! nor ceiling(n/2)!, but often coincides with one of them.

a(n) = k!, where k = floor(n/2) + d(n) and d = 0, 1, 2, ... . Below 1000, d = 1 arises 93 times, and d = 2 arises 4 times. See A056067 and A056068.

			For n = 10 or n = 11, floor(n/2)! = 5! = 120; 5!^2 = 14400 divides 10! = 14400*252 or 11! = 14400*2772. However, 10!/6!^2 = 7 and 11!/6!^2 = 77, i.e., (d + floor(n/2))^2 may divide n!. Here d = 1, but d > 1 also occurs as follows: for n = 416 or n = 417, floor(n/2) = 208, and 208!^2 divides 416! and 417!, but 209!^2 and 210!^2 also divide these factorials.

Cf. A000142, A001057, A001405, A055772, A056039, A056067, A056068, A105350.

a[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; (k-1)!]; Array[a, 33, 0] (* Amiram Eldar, May 24 2024 *)

a(n)^2 = A105350(n).

A056039 Largest k such that (k!)^2 divides n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37

Offset: 1

Labos Elemer, Jul 25 2000

nonn

Let d(n) = a(n) - floor(n/2). Then, d(n) >= 0, and below 1000, d=1 arises 93 times, and d=2 arises 4 times.

The first occurrence of d(k) = 0, 1, 2, ..., is at k = 2, 1 (=A056067(1)), 416 (=A056068(1)), 6950, 16348, 505930, ... . - Amiram Eldar, May 24 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A001057, A001405, A055772, A056038, A056067, A056068.

Cf. A001044, A105350.

a[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; k - 1]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

A105350(n) = A001044(a(n)).

A056042 a(n) = n!/(k!)^2, where k is the largest number such that (k!)^2 divides n!.

Original entry on oeis.org

1, 2, 6, 6, 30, 20, 140, 70, 630, 7, 77, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 178296, 5170584, 155117520, 4808643120, 601080390, 19835652870

Offset: 1

Labos Elemer, Jul 25 2000

nonn

Least integer of the form n!/{(n-k)!}^2.

Similar to but different from A001405.

			E.g. for n=9, 10, 11, 12, a(n)=630, 7, 77, 924 while the corresponding central binomial coefficients are 126, 252, 462, 924 respectively.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A000142, A001405, A000188, A055772, A096123, A096125.

f[n_] := Min[ Select[ Table[ n!/(n - k)!^2, {k, n}], IntegerQ[ # ] &]]; Table[ f[n], {n, 33}] (Robert G. Wilson v)

A056044 Let k be the largest number such that k^2 divides n! and let m be the largest number such that m!^2 divides n!; a(n) = k/m!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 10, 10, 30, 2, 2, 12, 12, 3, 3, 6, 30, 10, 10, 10, 30, 6, 6, 2, 2, 2, 30, 60, 60, 30, 210, 42, 42, 42, 42, 1, 1, 2, 2, 4, 4, 4, 4, 4, 84, 21, 21, 14, 14, 14, 42, 6, 6, 2, 2, 2, 10, 10, 70, 140, 140, 14, 126, 3, 3, 6, 30

Offset: 1

Labos Elemer, Jul 25 2000

nonn

			For n = 11, 11! = 6! * 6! * 77, so A000188(11!) = A056038(11) = 6! and a(11) = 6!/6! = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000188, A055772, A001405, A001057, A056038.

f[p_, e_] := p^Floor[e/2]; b[1] = 1; b[n_] := Times @@ f @@@ FactorInteger[n!];
c[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; (k-1)!];
a[n_] := b[n] / c[n]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

a(n) = A000188(n!)/A056038(n).

Name corrected by Amiram Eldar, May 24 2024

A056627 a(n) = A056622(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 12, 12, 36, 720, 720, 480, 480, 1680, 3024, 12096, 12096, 145152, 145152, 7257600, 345600, 1900800, 1900800, 136857600, 684288000, 4447872000, 4447872000, 435891456000, 435891456000, 3138418483200, 3138418483200, 6276836966400, 190207180800

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name "Square root of largest unitary square divisor of n!" was incorrect. See A374989 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

			a(12) = A056622(12!) = A000188(12!)/A055229(12!) = 1440/3 = 480.

Cf. A055772, A055230, A000188, A055229, A056622, A056628, A374989.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 33] (* Amiram Eldar, Jul 08 2024 *)

A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017
a(n) = sqrtint(A056623(n!)); \\ Michel Marcus, Aug 16 2020

a(n) = A055772(n)/A055230(n) = A000188(n!)/A055229(n!).
a(n) = A056622(n!). - Michel Marcus, Aug 16 2020
a(n) = sqrt(A056628(n)). - Amiram Eldar, Jul 08 2024

More terms from Michel Marcus, Aug 16 2020

Incorrect name replaced with a formula by Amiram Eldar, Jul 26 2024

A065887 Smallest number whose square is divisible by n!.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 60, 420, 1680, 5040, 5040, 55440, 332640, 4324320, 8648640, 43243200, 172972800, 2940537600, 8821612800, 167610643200, 335221286400, 7039647014400, 14079294028800, 323823762662400, 647647525324800, 3238237626624000, 6476475253248000

Offset: 0

Henry Bottomley, Nov 27 2001

nonn

			a(10) = 5040 since 10! = 3628800 and the smallest square divisible by this is 25401600 = 3628800*7 = 5040^2.

Alois P. Heinz, Table of n, a(n) for n = 0..735 (first 101 terms from Kevin P. Thompson)

Cf. A000142, A019554, A055772, A065886.

a:= n-> mul(i[1]^ceil(i[2]/2), i=ifactors(n!)[2]):
seq(a(n), n=0..26);  # Alois P. Heinz, Jan 24 2022

f[p_, e_] := p^Ceiling[e/2]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30, 0] (* Amiram Eldar, Feb 11 2024 *)

a(n) = A019554(A000142(n)) = sqrt(A065886(n)) = A000142(n)/A055772(n).

Missing a(0) inserted, formula corrected, and a(25)-a(26) added by Kevin P. Thompson, Jan 24 2022

cp's OEIS Frontend

A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).

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A055204 Squarefree part of n!: n! divided by its largest square divisor.

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A055071 Largest square dividing n!.

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A055993 Number of square divisors of n!.

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A056038 Largest factorial k! such that (k!)^2 divides n!.

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A056039 Largest k such that (k!)^2 divides n!.

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A056042 a(n) = n!/(k!)^2, where k is the largest number such that (k!)^2 divides n!.

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