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A180000 a(n) = lcm{1,2,...,n} / swinging_factorial(n) = A003418(n) / A056040(n). 6
1, 1, 1, 1, 2, 2, 3, 3, 12, 4, 10, 10, 30, 30, 105, 7, 56, 56, 252, 252, 1260, 60, 330, 330, 1980, 396, 2574, 286, 2002, 2002, 15015, 15015, 240240, 7280, 61880, 1768, 15912, 15912, 151164, 3876, 38760, 38760, 406980, 406980, 4476780, 99484, 1144066 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Characterization: Let e_{p}(m) denote the exponent of the prime p in the prime factorization of m and [.] denote the Iverson bracket, then

e_{p}(a(n)) = Sum_{k>=1} [floor(n/p^k) is even].

This implies, among other things, that no prime > floor(n/2) can divide a(n). The prime exponents e_{2}(a(2n)) give Guy Steele's sequence GS(5,3) A080100.

Asymptotics: log a(n) ~ n(1 - log 2). It is conjectured that log a(n) ~ n(1 - log 2) + O(n^{1/2+eps}) for all eps > 0.

Bounds: A056040(floor(n/3)) <= a(n) <= A056040(floor(n/2)) if n >= 285.

REFERENCES

Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.

LINKS

Peter Luschny, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = 2^(-n)*Product_{1<=k<=n} A014963(k)*(k/2)^((-1)^k).

MAPLE

a := proc(n) local A014963, k;

A014963 := proc(n) if n < 2 then 1 else numtheory[factorset](n);

if 1 < nops(%) then 1 else op(%) fi fi end;

mul(A014963(k)*(k/2)^((-1)^k), k=1..n)/2^n end;

# Also:

A180000 := proc(n) local lcm, sf;

lcm := ilcm(seq(i, i=1..n));

sf := n!/iquo(n, 2)!^2;

lcm/sf end;

MATHEMATICA

a[0] = 1; a[n_] := LCM @@ Range[n] / (n! / Floor[n/2]!^2); Table[a[n], {n, 0, 46}] (* Jean-Fran├žois Alcover, Jul 23 2013 *)

PROG

(PARI) L=1; X(n)={ ispower(n, , &n); if(isprime(n), n, 1); }

Y(n)={ a=X(n); b=if(bitand(1, n), a, a*(n/2)^2); L=(b*L)/n; }

A180000_list(n)={ L=1; vector(n, m, Y(m)); }  \\ for n>0

(Sage)

def Exp(m, n) :

    s = 0; p = m; q = n//p

    while q > 0 :

        if is_even(q) :

            s = s + 1

        p = p * m

        q = n//p

    return s

def A180000(n) :

    A = [1, 1, 1, 1, 2, 2, 3, 3, 12]

    if n < 9 : return A[n]

    R = []; r = isqrt(n)

    P = Primes(); p = P.first()

    while p <= n//2 :

        if p <= r : R.append(p^Exp(p, n))

        elif p <= n//3 :

            if is_even(n//p) : R.append(p)

        else : R.append(p)

        p = P.next(p)

    return mul(x for x in R)

CROSSREFS

Cf. A003418, A014963, A056040.

Sequence in context: A097365 A156906 A216245 * A053094 A196080 A124516

Adjacent sequences:  A179997 A179998 A179999 * A180001 A180002 A180003

KEYWORD

nonn

AUTHOR

Peter Luschny, Aug 17 2010

STATUS

approved

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Last modified December 8 17:36 EST 2019. Contains 329865 sequences. (Running on oeis4.)