A055773 -id:A055773 - cp's OEIS frontend

A333072 Least k such that Sum_{i=1..n} k^i / i is a positive integer.

1, 2, 6, 6, 30, 10, 70, 70, 210, 168, 1848, 1848, 18018, 8580, 2574, 2574, 102102, 102102, 831402, 2771340, 3233230, 587860, 43266496, 117630786, 162249360, 145088370, 145088370, 2897310, 672175920, 672175920, 18232771830, 18232771830, 44279588730, 8886561060

Offset: 1

Jinyuan Wang, Mar 10 2020

nonn

Note that the denominator of (Sum_{i=1..n} k^i/i) - k^p/p can never be divisible by p, where n/2 < p <= n. Therefore, for the expression to be an integer, such p must divide k. Thus, a(n) = k is divisible by A055773(n).

Chai Wah Wu, Table of n, a(n) for n = 1..61

Cf. A034386, A055773, A332734, A333196.

a(n) = {my(m = prod(i=primepi(n/2)+1, primepi(n), prime(i)), k = m); while (denominator(sum(i=2, n, k^i/i)) != 1, k += m); k; }

from sympy import primorial, lcm
def A333072(n):
    f = 1
    for i in range(1,n+1):
        f = lcm(f,i)
    f, glist = int(f), []
    for i in range(1,n+1):
        glist.append(f//i)
    m = 1 if n < 2 else primorial(n,nth=False)//primorial(n//2,nth=False)
    k = m
    while True:
        p,ki = 0, k
        for i in range(1,n+1):
            p = (p+ki*glist[i-1]) % f
            ki = (k*ki) % f
        if p == 0:
            return k
        k += m # Chai Wah Wu, Apr 04 2020

a(n) <= A034386(n).

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

A056195 a(n) = n! divided by its characteristic cube divisor A056194.

Original entry on oeis.org

1, 2, 6, 3, 15, 720, 5040, 5040, 45360, 3628800, 39916800, 17740800, 230630400, 403603200, 1307674368, 20922789888, 355687428096, 51218989645824, 973160803270656, 2432902008176640000, 5516784599040000

Offset: 1

Labos Elemer, Aug 02 2000

nonn

			n = 10, a(10) = 10! because g(10!) = 1.
n = 9, a(9) = 45320 because 9! = 2*2*2*2*2*2*2*3*3*5*7 and g(9!) = 2, so a(9) = 9!/8.

Amiram Eldar, Table of n, a(n) for n = 1..491

Cf. A055229, A055230, A055071, A055772, A055773.

f[p_, e_] := If[OddQ[e] && e > 1, p^3, 1]; a[n_] := n! / (Times @@ f @@@ FactorInteger[n!]); Array[a, 20] (* Amiram Eldar, Sep 06 2020 *)

The CCD of n! is the cube of g = A055229(n!) = A055230(n). So a(n) = n!/ggg = L*L*f where L = A000188(n!)/A055229(n!) = A055772(n)/A055230(n) and f = A055231(n!) = A055773(n).

A193477 Denominator(n!/floor(n/2)!^4).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 9, 9, 288, 32, 400, 400, 43200, 43200, 1058400, 70560, 18063360, 18063360, 6584094720, 6584094720, 3292047360000, 156764160000, 9484231680000, 9484231680000, 8194376171520000, 327775046860800, 27696991459737600, 1025814498508800

Offset: 0

Peter Luschny, Jul 30 2011

nonn

Cf. A055773.

Maple
```
A193477 := n -> denom(n!/iquo(n,2)!^4);
```

a(n) = Denominator(A056040(n)^2/n!).

a(n) = Denominator(A056040(n)/floor(n/2)!^2).

A333073 Least k such that Sum_{i=1..n} (-k)^i / i is a positive integer.

Original entry on oeis.org

1, 2, 6, 6, 30, 20, 140, 140, 210, 42, 462, 462, 12012, 3432, 6006, 6006, 87516, 87516, 1108536, 3048474, 2586584, 2586584, 44618574, 44618574, 60843510, 17160990, 17160990, 14263680, 782050830, 782050830, 3806842470, 3806842470, 16830250920, 16830250920

Offset: 1

Jinyuan Wang, Mar 31 2020

nonn

Note that the denominator of (Sum_{i=1..n} (-k)^i/i) - (-k)^p/p can never be divisible by p, where n/2 < p <= n. Therefore, for the expression to be an integer, such p must divide k. Thus, a(n) = k is divisible by A055773(n).

Chai Wah Wu, Table of n, a(n) for n = 1..60

Cf. A034386, A055773, A333072, A333074.

a(n) = {my(m = prod(i=primepi(n/2)+1, primepi(n), prime(i)), k = m); while(denominator(sum(i=2, n, (-k)^i/i)) != 1, k += m); k; }

from sympy import primorial, lcm
def A333073(n):
    f = 1
    for i in range(1,n+1):
        f = lcm(f,i)
    f = int(f)
    glist = []
    for i in range(1,n+1):
        glist.append(f//i)
    m = 1 if n < 2 else primorial(n,nth=False)//primorial(n//2,nth=False)
    k = m
    while True:
        p,ki = 0, -k
        for i in range(1,n+1):
            p = (p+ki*glist[i-1]) % f
            ki = (-k*ki) % f
        if p == 0:
            return k
        k += m # Chai Wah Wu, Apr 02 2020

a(n) <= A034386(n).

A161887 A product of quotients of factorials.

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 840, 7560, 15120, 110880, 166320, 1441440, 2882880, 10810800, 43243200, 183783600, 367567200, 2793510720, 6983776800, 58663725120, 117327450240, 299836817280, 2698531355520, 7495920432000

Offset: 1

Peter Luschny, Jun 21 2009

Definition: Let b(n,k) = floor(n/2^k)! and m = log[2](n) then c(n) = product_{k=1..m} b(n,k) / b(n,k+1)^2.
a(n) is the sequence derived from c(n) by eliminating duplicates and sorting the values.
a(1) through a(19) are highly composite numbers (A002182).
The number of divisors of a(1) through a(28) are number of divisors of highly composite numbers (A002183).
A055773(floor(n/2)) is a divisor of a(n), a(n)/A055773(floor(n/2)) after eliminating duplicates and sorting starts 1,4,24,216,1440,2160,..

Amiram Eldar, Table of n, a(n) for n = 1..1669

Cf. A002182, A002183.

a := proc(n) local m,k; m := nops(convert(n,base,2));
mul(iquo(n,2^k)!/iquo(n,2^(k+1))!^2,k=1..m-1) end:
seq(a(i),i=1..50): A:=sort(convert({%},list));

b[n_, k_] := Floor[n/2^k]!; c[n_] := Product[b[n, k]/b[n, k+1]^2, {k, 1, Log[2, n]}]; A = Array[c, 50] // DeleteDuplicates // Sort (* Jean-François Alcover, Jun 17 2019 *)

A220027 a(n) = product(i >= 0, P(n, i)^(2^i)) where P(n, i) = product(p prime, n/2^(i+1) < p <= n/2^i).

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 180, 1260, 5040, 5040, 25200, 277200, 2494800, 32432400, 227026800, 227026800, 3632428800, 61751289600, 61751289600, 1173274502400, 29331862560000, 29331862560000, 322650488160000, 7420961227680000, 601097859442080000, 601097859442080000

Offset: 0

Peter Luschny, Mar 30 2013

nonn

a(n) are the partial products of A219964(n).
a(n) divides n!, n!/a(n) = 1, 1, 1, 1, 2, 2, 4, 4, 8, 72, 144, 144, 192...
The swinging factorial (A056040) divides a(n), a(n)/n$ = 1, 1, 1, 1, 2,...
The primorial of n (A034386) divides a(n), a(n)/n# = 1, 1, 1, 1, 2, 2, 6,..
If p^k is the largest power of a prime dividing a(n) then k is 2^n for some n >= 0.
a(n) / A055773(n) is the largest square dividing a(n), a(n) / A055773(n) = A008833(a(n)).

Cf. A055773.

a := proc(n) local k; `if`(n < 2, 1,
mul(k, k = select(isprime, [$iquo(n, 2)+1..n]))*a(iquo(n,2))^2) end:
seq(a(i), i=0..25);

def a(n) :
    if n < 2 : return 1
    return mul(k for k in prime_range(n//2+1,n+1))*a(n//2)^2
[a(n) for n in (0..25)]

A370971 A008336(n) is divisible by the product of the primes p such that n/2 <= p < n; a(n) is the quotient.

Original entry on oeis.org

1, 1, 1, 1, 8, 8, 4, 4, 32, 288, 144, 144, 12, 12, 6, 90, 1440, 1440, 80, 80, 4, 84, 42, 42, 1008, 25200, 12600, 340200, 12150, 12150, 405, 405, 12960, 427680, 213840, 7484400, 207900, 207900, 103950, 4054050, 162162000, 162162000, 3861000, 3861000, 87750, 1950, 975, 975, 46800, 2293200, 45864, 2339064, 44982

Offset: 1

N. J. A. Sloane, Apr 13 2024

nonn

			For n= 7, A008336(7) = 20. The only prime with 3.5 <= p < 7 is 5, so a(7) = 20/5 = 4.
For n= 8, A008336(7) = 140. The only primes with 4 <= p < 8 are 5 and 7, so a(8) = 140/(5*7) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..2732

Cf. A008336, A055773 (note that A055773 has offset 0).

a(n) = A008336(n)/A055773(n-1).

A111866 Divide n! repeatedly by i! for i from floor(n/2) down through 2; a(n) = remaining quotient.

Original entry on oeis.org

1, 1, 2, 6, 3, 15, 5, 35, 35, 105, 7, 77, 77, 1001, 143, 429, 2145, 36465, 12155, 230945, 46189, 323323, 29393, 676039, 676039, 16900975, 1300075, 2340135, 334305, 9694845, 215441, 6678671, 100180065, 3305942145, 64822395, 2268783825

Offset: 0

Jon Perry, Nov 23 2005

nonn

			a(9): 9!/(4!*4!*3!) = 105.

Cf. A055773.

f[n_] := Block[{k = Floor[n/2], m = n!}, While[k > 1, While[ Mod[m, k! ] == 0, m /= k! ]; k-- ]; m]; Table[ f[n], {n, 0, 35}] (* Robert G. Wilson v *)

a(n)=c=n!;forstep(d=floor(n/2),2,-1, while (c%(d!)==0,c/=d!));c

More terms from Robert G. Wilson v, Nov 25 2005

cp's OEIS Frontend

A333072 Least k such that Sum_{i=1..n} k^i / i is a positive integer.

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

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A056195 a(n) = n! divided by its characteristic cube divisor A056194.

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A193477 Denominator(n!/floor(n/2)!^4).

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A333073 Least k such that Sum_{i=1..n} (-k)^i / i is a positive integer.

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PARI

Python

Formula

A161887 A product of quotients of factorials.

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Maple

Mathematica

A220027 a(n) = product(i >= 0, P(n, i)^(2^i)) where P(n, i) = product(p prime, n/2^(i+1) < p <= n/2^i).

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Sage

A370971 A008336(n) is divisible by the product of the primes p such that n/2 <= p < n; a(n) is the quotient.

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Formula

A111866 Divide n! repeatedly by i! for i from floor(n/2) down through 2; a(n) = remaining quotient.

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