A056040 -id:A056040 - cp's OEIS frontend

A163640 The radical of the swinging factorial A056040 for odd indices.

1, 6, 30, 70, 210, 462, 6006, 4290, 72930, 461890, 1939938, 4056234, 6760390, 1560090, 6463230, 200360130, 2203961430, 907513530, 33578000610, 22974421470, 941951280270, 5786272150230, 526024740930, 1074920122770, 7524440859390, 25583098921926, 104300326374006, 1912172650190110

Offset: 0

Peter Luschny, Aug 02 2009

nonn

Let $ denote the swinging factorial. a(n) is the radical of (2*n+1)$ which is the product of the prime numbers dividing (2*n+1)$. It is the largest squarefree divisor of (2*n+1)$, and so also described as the squarefree kernel of (2*n+1)$.

			(2*5+1)$ = 2772 = 2^2*3^2*7*11. Therefore a(5) = 2*3*7*11 = 462.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

A056040(n) = n$, A163641(n) = rad(n$), A080397(n) = rad((2n)$).

a := proc(n) local p; mul(p,p=numtheory[factorset]((2*n+1)!/iquo(2*n+1,2)!^2)) end:

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Times @@ FactorInteger[sf[2*n + 1]][[All, 1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 30 2013 *)

More terms from Michel Marcus, Aug 22 2025

A163644 Product of primes which do not exceed n and do not divide the swinging factorial n$ (A056040).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 5, 5, 35, 7, 7, 7, 21, 21, 105, 5, 55, 55, 165, 33, 429, 143, 1001, 1001, 1001, 1001, 1001, 91, 1547, 221, 221, 221, 4199, 323, 323, 323, 2261, 2261, 24871, 24871, 572033, 572033, 572033, 81719, 408595, 24035, 312455

Offset: 0

Peter Luschny, Aug 02 2009

nonn

			a(20) = 105 because in the prime-factorization of 20$ the primes 3, 5 and 7 are missing and 3*5*7 = 105.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A056040, A034386, A163641, A056610.

a := proc(n) local p; mul(p,p=select(isprime,{$1..n})
minus numtheory[factorset](n!/iquo(n,2)!^2)) end:

A034386[x_] := Apply[Times, Table[Prime[w], {w, 1, PrimePi[x]}]];
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f + 1, n - f]/f!];
A163641[0] = 1; A163641[n_] := Times @@ FactorInteger[sf[n]][[All, 1]]; Join[{1}, Table[A034386[n]/A163641[n], {n, 1, 50}]] (* G. C. Greubel, Aug 01 2017 *)

a(n) = primorial(n) / rad(n$) = A034386(n) / A163641(n).

A182910 Number of unitary prime divisors of the swinging factorial (A056040) n$ = n! / floor(n/2)!^2.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 3, 3, 4, 5, 4, 5, 4, 6, 5, 6, 5, 5, 4, 4, 3, 4, 5, 6, 7, 8, 6, 6, 7, 8, 7, 7, 8, 9, 9, 10, 9, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 10, 11, 13, 12, 13, 11, 12, 11, 10, 11, 13, 12, 13

Offset: 0

Peter Luschny, Mar 14 2011

nonn

A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. A unitary prime divisor of the swinging factorial n$ can be smaller than n/2. For n >= 30 the swinging factorial has more unitary prime divisors than the factorial and it never has fewer unitary prime divisors. Thus a(n) >= PrimePi(n) - PrimePi(n/2).

			16$ = 2*3*3*5*11*13. So 16$ has one non-unitary prime divisor and a(16) = 4.

Cf. A056171.

UnitaryPrimeDivisor := proc(f,n) local k, F; F := f(n):
add(`if`(igcd(iquo(F,k),k)=1,1,0),k=numtheory[factorset](F)) end;
A056040 := n -> n!/iquo(n,2)!^2;
A182910 := n -> UnitaryPrimeDivisor(A056040,n);
seq(A182910(i), i=1..LEN);

Table[Function[m, If[m == 1, 0, Count[FactorInteger[m][[All, -1]], 1]]][n!/Floor[n/2]!^2], {n, 0, 67}] (* Michael De Vlieger, Aug 02 2017 *)

from sympy import factorint, factorial
def a056169(n): return 0 if n==1 else sum(1 for i in factorint(n).values() if i==1)
def a056040(n): return factorial(n)//factorial(n//2)**2
def a(n): return a056169(a056040(n))
print([a(n) for n in range(68)]) # Indranil Ghosh, Aug 02 2017

A182921 Sum of exponents in prime-power factorization of the swinging factorial (A056040) n$ = n!/floor(n/2)!^2; also bigomega(n$).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 4, 3, 5, 5, 6, 5, 6, 6, 8, 6, 7, 6, 7, 6, 8, 8, 9, 7, 9, 9, 12, 11, 12, 11, 12, 9, 11, 11, 13, 11, 12, 12, 14, 12, 13, 12, 13, 12, 15, 15, 16, 13, 15, 14, 16, 15, 16, 14, 16, 14, 16, 16, 17, 15, 16, 16, 19, 15, 17, 16, 17, 16, 18, 17, 18, 15

Offset: 0

Peter Luschny, Mar 14 2011

nonn

			16$ = 2*3*3*5*11*13. Thus a(16) = 6.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A022559, A056040.

A056040 := n -> n! / iquo(n,2)!^2;
A182921 := n -> numtheory[bigomega](A056040(n)): seq(A182921(i), i=0..70);

a[n_] := PrimeOmega[n!/Quotient[n, 2]!^2];
Table[a[n], {n, 0, 64}] (* Jean-François Alcover, Jun 18 2019 *)

a(n) = bigomega(n! / (n\2)!^2); \\ Amiram Eldar, Jun 13 2025

A182923 a(n) = n$ / A055773(n), where n$ denotes the swinging factorial (A056040).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 2, 18, 36, 36, 12, 12, 24, 360, 90, 90, 20, 20, 4, 84, 168, 168, 28, 700, 1400, 37800, 5400, 5400, 720, 720, 90, 2970, 5940, 207900, 23100, 23100, 46200, 1801800, 180180, 180180, 17160

Offset: 0

Peter Luschny, Mar 05 2011

nonn

a(n) = n$ * P(floor(n/2))/P(n), P(n) primorial number A034386.

A182922(n) / a(n) = A000142(n) / A056040(n) = A180064(n).

Cf. A182922, A034386, A055773, A180064.

swingfact := n -> n! / iquo(n,2)!^2;
A182923 := n -> swingfact(n) / mul(k, k=select(isprime, [$iquo(n,2)+1..n])):

sf[n_] := n!/Floor[n/2]!^2;
a[n_] := sf[n]/Numerator[n!/Floor[n/2]!^4];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jun 22 2019 *)

A190905 Euler transform of the swinging factorial A056040.

Original entry on oeis.org

1, 1, 3, 9, 18, 60, 117, 371, 747, 2199, 4697, 12735, 28571, 72815, 169176, 412440, 978086, 2316754, 5547293, 12909723, 30966639, 71357601, 170636159, 391242623, 930120982, 2128073530, 5023630830, 11486060090, 26918052717, 61539213693, 143227189518

Offset: 0

Peter Luschny, Jul 06 2011

nonn

M. Bernstein and N. J. A. Sloane, Some Canonical Integer Sequences, (arXiv:0205301v1), May 28 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A107895.

EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
`if`(n=0,1,add(add(d*p(d),d=numtheory[divisors](j))*b(n-j),j=1..n)/n) end end:
A190905 := EulerTrans(n->n!/iquo(n,2)!^2): seq( A190905(n),n=0..30); # After Alois P. Heinz, A000335.

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; EulerTrans[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = EulerTrans[sf]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 29 2013, after Maple *)

A246663 Products of swinging factorials A056040.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 30, 32, 36, 40, 48, 60, 64, 70, 72, 80, 96, 120, 128, 140, 144, 160, 180, 192, 216, 240, 252, 256, 280, 288, 320, 360, 384, 400, 420, 432, 480, 504, 512, 560, 576, 600, 630, 640, 720, 768, 800, 840, 864, 900, 960, 1008, 1024

Offset: 1

Peter Luschny, Sep 09 2014

Cf. A001013 is a sublist.

sw[n_] := n!/(Floor[n/2]!)^2; lim = 40; For[p = 0; a = f = Table[sw[n], {n, lim}], p =!= a, p = a; a = Select[Union@@Outer[Times, f, a], #<= sw[lim]&]]; a (* Hans Havermann, Sep 09 2014 *)

# For example prod_hull(A008578) are the natural numbers.
def prod_hull(f, K):
    S = []; newS = []
    n = 0
    while f(n) <= K:
        newS.append(f(n))
        n += 1
    while newS != S:
        S = newS; T = []
        for s in S:
            M = map(lambda n: n*s , S)
            T.extend(filter(lambda n: n <= K, M))
        newS = Set(T).union(Set(S))
    return sorted(newS)
prod_hull(lambda n: factorial(n)/factorial(n//2)^2, 1024)

A261129 Highest exponent in prime factorization of the swinging factorial (A056040).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 2, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 5, 5, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 2, 4, 4, 4, 3, 3, 4, 4, 4

Offset: 2

Peter Luschny, Oct 31 2015

nonn

A263922 is a subsequence.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A000120, A000523, A056040, A263922.

swing := n -> n!/iquo(n,2)!^2:
max_exp := n -> max(seq(s[2], s=ifactors(n)[2])):
seq(max_exp(swing(n)), n=2..88);

a[n_] := Max[FactorInteger[n!/Quotient[n, 2]!^2][[;; , 2]]]; Array[a, 100, 2] (* Amiram Eldar, Jul 29 2023 *)

swing = lambda n: factorial(n)//factorial(n//2)^2
max_exp = lambda n: max(e for p, e in n.factor())
[max_exp(swing(n)) for n in (2..88)]

a(n) = A051903(A056040(n)) for n>=2.

A000120(floor(n/2)) <= a(n) <= A000523(n), (n>=2).

A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457), read by rows.

Original entry on oeis.org

1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790

Offset: 0

Peter Luschny, Aug 07 2009

			Triangle begins:
   1;
  -1,   6;
   1, -12,   30;
  -1,  18,  -90,   140;
   1, -24,  180,  -560,   630;
  -1,  30, -300,  1400, -3150,   2772;
   1, -36,  450, -2800,  9450, -16632, 12012;

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Row sums are the inverse binomial transform of the beta numbers (A163872).

Cf. A163649, A098473, A056040.

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n, k) (-1)^(n-k)*binomial(n,k)*swing(2*k+1) end:
seq(print(seq(a(n,k),k=0..n)),n=0..8);

T[n_,k_] := ((-1)^(Mod[k,2]+n)*((2*k+1)!/(k!)^2)*Binomial[n,n-k]);
Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Oct 07 2023 *)

For n >= 0, k >= 0, T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ where i$ denotes the swinging factorial of i (A056040).
Conjectural g.f.: sqrt(1 + t)/(1 + (1 - 4*x)*t)^(3/2) = 1 + (-1 + 6*x)*t + (1 - 12*x + 30*x^2)*t^2 + .... - Peter Bala, Nov 10 2013
T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - Detlef Meya, Oct 07 2023

A294038 a(n) = Product_{k=0..n-1} n/gcd(n, A056040(k)).

Original entry on oeis.org

1, 1, 4, 27, 64, 3125, 108, 823543, 65536, 14348907, 12500, 285311670611, 746496, 302875106592253, 3294172, 7119140625, 1099511627776, 827240261886336764177, 1549681956, 1978419655660313589123979, 40000000000, 13349464742886867, 1141246682444

Offset: 0

Peter Luschny, Nov 05 2017

nonn

Cf. A056040, A051674, A058067.

seq(mul(n/igcd(n, A056040(k)), k=0..n-1), n=0..22);

cp's OEIS Frontend

A163640 The radical of the swinging factorial A056040 for odd indices.

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A163644 Product of primes which do not exceed n and do not divide the swinging factorial n$ (A056040).

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A182910 Number of unitary prime divisors of the swinging factorial (A056040) n$ = n! / floor(n/2)!^2.

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A182921 Sum of exponents in prime-power factorization of the swinging factorial (A056040) n$ = n!/floor(n/2)!^2; also bigomega(n$).

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A182923 a(n) = n$ / A055773(n), where n$ denotes the swinging factorial (A056040).

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A190905 Euler transform of the swinging factorial A056040.

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A246663 Products of swinging factorials A056040.

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A261129 Highest exponent in prime factorization of the swinging factorial (A056040).

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