A048148 -id:A048148 - cp's OEIS frontend

A049614 n! divided by its squarefree kernel.

1, 1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 2903040, 43545600, 696729600, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 115880067072000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000

Offset: 0

Labos Elemer

nonn

Also product of composite numbers less than or equal to n. - Benoit Cloitre, Aug 18 2002
Also n! divided by n primorial (or n!/n#). - Cino Hilliard, Mar 26 2006
From Alexander R. Povolotsky and Peter J. C. Moses, Aug 27 2007: (Start)
It appears that a(n) = smallest positive number m such that the sequence b(n) = { m (i^1 + 1!) (i^2 + 2!) ... (i^n + n!) / n! : i >= 0 } takes integral values. [It would be nice to have a proof of this! - N. J. A. Sloane] Cf. A064808 (for n=2), A131682 (for n=3), A131683 (for n=4), A131527 (for n=5), A131684 (for n=6), A131528. See also A129995, A131685. (End)
It appears that every term > 4 is divisible by 24. - Alexander R. Povolotsky, Oct 18 2007
The above comment is correct since each term divides the next. - Charles R Greathouse IV, Jan 16 2012
When n is not a prime number, then a(n)=m*n, where m is some integer >0; such a(n) make up the A036691 Otherwise, when n is a prime number, then a(n)=a(k), where k is the largest nonprime number preceding n (kAlexander R. Povolotsky, Aug 21 2012

			n = 11: 11! = 39916800 = 2310*17280 and 2310=2*3*5*7*11.

Cf. A000142, A002110, A003418, A034386, A036691, A045948, A048148.

A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
[A049614(n): n in [0..40]]; // G. C. Greubel, Jul 21 2023

primorial := n -> mul(k, k=select(isprime, [$1..n]));
A049614 := n -> factorial(n)/primorial(n);
seq(A049614(i),i=0..24); # Peter Luschny, Feb 16 2013

Table[n!/Product[ Prime[i], {i, PrimePi[n]}], {n, 24}]

PARI
```
a(n)=prod(i=1,n,i^if(isprime(i),0,1))
```

a(n)=n!/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Aug 30 2012

def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1,1+prime_pi(n)))
[A049614(n) for n in range(41)] # G. C. Greubel, Jul 21 2023

a(n) = A000142(n)/A034386(n).

Edited by N. J. A. Sloane, Oct 07 2007

Offset set to 0, a(0)=1 prepended to data, Peter Luschny, Feb 16 2013

A045948 a(n) = A003418(n)/A034386(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 12, 12, 12, 12, 12, 12, 12, 24, 24, 24, 24, 24, 24, 24, 24, 24, 120, 120, 360, 360, 360, 360, 360, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040

Offset: 1

Labos Elemer

nonn

			n=11: lcm(1..11) = 27720 = 8*9*5*7*11 = 2310*12. A034386(11)=2310, so the quotient is 12. Thus a(11) = 12.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A003418, A034386, A002110, A000142, A049536, A048148.

Table[Exp[Sum[MangoldtLambda[n], {n, 1, m}]]/ Product[x, {x, Prime[Range[PrimePi[m]]]}], {m, 1, 57}] (* Fred Daniel Kline, Apr 02 2015 *)

a(n)=lcm([1..n])/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Apr 02 2015; corrected by Michel Marcus, Dec 26 2020

A025476 Prime root of n-th nontrivial prime power (A025475, A246547).

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 2, 7, 2, 3, 11, 5, 2, 13, 3, 2, 17, 7, 19, 2, 23, 5, 3, 29, 31, 2, 11, 37, 41, 43, 2, 3, 13, 47, 7, 53, 5, 59, 61, 2, 67, 17, 71, 73, 79, 3, 19, 83, 89, 2, 97, 101, 103, 107, 109, 23, 113, 11, 5, 127, 2, 7, 131, 137, 139, 3, 149, 151, 29, 157, 163, 167, 13, 31, 173, 179

Offset: 1

David W. Wilson

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

Cf. A025473, A025475, A048148, A246547.

cvm := proc(n, level) local f,opf; if n < 2 then RETURN() fi;
f := ifactors(n); opf := op(1,op(2,f)); if nops(op(2,f)) > 1 or
op(2,opf) <= level then RETURN() fi; op(1,opf) end:
A025476_list := n -> seq(cvm(i,1),i=1..n); # n is search limit
A025476_list(30000);  # Peter Luschny, Sep 21 2011
# Alternative:
isA246547 := n -> n > 1 and not isprime(n) and type(n, 'primepower'):
seq(ifactors(p)[2][1][1], p in select(isA246547, [$1..30000])); # Peter Luschny, Jul 15 2023

Transpose[ Flatten[ FactorInteger[ Select[ Range[2, 30000], !PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ]] == 0 &]], 1]][[1]] (* Robert G. Wilson v *)

forcomposite(n=4,10^5,if( ispower(n, , &p) && isprime(p), print1(p,", "))) \\ Joerg Arndt, Sep 11 2021

from sympy import primepi, integer_nthroot, primefactors
def A025476(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return primefactors(kmax)[0] # Chai Wah Wu, Aug 15 2024

cp's OEIS Frontend

A049614 n! divided by its squarefree kernel.

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A045948 a(n) = A003418(n)/A034386(n).

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A025476 Prime root of n-th nontrivial prime power (A025475, A246547).

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