A045948 -id:A045948 - cp's OEIS frontend

A048148 Distinct elements of A045948.

1, 2, 4, 12, 24, 120, 360, 720, 5040, 10080, 30240, 332640, 1663200, 3326400, 43243200, 129729600, 259459200, 4410806400, 30875644800, 586637251200, 1173274502400, 26985313555200, 134926567776000, 404779703328000, 11738611396512000, 363896953291872000

Offset: 1

Labos Elemer

nonn

A001222(a(n)) = n - 1. - Eric Desbiaux, Apr 05 2018

Chai Wah Wu, Table of n, a(n) for n = 1..412 (terms 1..237 from Michael De Vlieger)

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

Map[Function[m, Exp[Sum[MangoldtLambda@ n, {n, m}]]/Product[x, {x, Prime@ Range@ PrimePi@ m}]], Select[Range[10^3], Or[# == 1, And[PrimePowerQ@ #, ! PrimeQ@ #]] &] ] (* Michael De Vlieger, Aug 01 2017, after Fred Daniel Kline at A045948 *)

# uses program from A025476
from math import prod
def A048148(n): return prod(A025476(i) for i in range(1,n)) # Chai Wah Wu, Aug 15 2024

A025476(n) = a(n+1) / a(n). - Eric Desbiaux, Jun 22 2013

More terms from James Sellers, May 03 2000

More terms from Michael De Vlieger, Aug 01 2017

A049614 n! divided by its squarefree kernel.

Original entry on oeis.org

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

A056813 Largest non-unitary prime factor of LCM(1,...,n); that is, the largest prime which occurs to power > 1 in prime factorization of LCM(1,..,n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Labos Elemer, Aug 28 2000

For n>0, prime(n) appears {(prime(n+1))^2 - (prime(n))^2} times [from n=(prime(n))^2 to n=(prime(n+1))^2 - 1], that is, A000040(n) appears A069482(n) times (from n=A001248(n) to n=A084920(n+1)). - Lekraj Beedassy, Mar 31 2005
a(n) is the largest prime factor of A045948(n). [Matthew Vandermast, Oct 29 2008]
Alternative definition: a(n) = largest prime <= sqrt(n) (considering 1 as prime for this occasion, see A008578 for the 19th century definition of primes). - Jean-Christophe Hervé, Oct 29 2013

			The j-th prime appears at the position of its square, at n = prime(j)^2.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Cf. A054041, A056170.

Cf. A000040, A001248, A008578, A069482.

Table[f = Transpose[FactorInteger[LCM @@ Range[n]]]; pos = Position[f[[2]], ?(# > 1 &)]; If[pos == {}, 1, f[[1, pos[[-1]]]][[1]]], {n, 100}] (* _T. D. Noe, Oct 30 2013 *)

a(n) = prime(w) if prime(w)^2 <= n < prime(w+1)^2.

To get the sequence, repeat 1 three times, and then for any k >= 1, repeat A000040(k) A069482(k) times; or equivalently, for any k >= 1, repeat A008578(k) a number of times equal to A008578(k+1)^2 - A008578(k)^2. - Jean-Christophe Hervé, Oct 29 2013

Corrected offset by Jean-Christophe Hervé, Oct 29 2013

A168262 Intersection of A003418 and A116998.

Original entry on oeis.org

1, 2, 6, 12, 60, 420, 840, 27720, 360360, 5354228880

Offset: 1

Matthew Vandermast, Nov 23 2009

If, for some prime p, A045948(p) > p^2, then all members of the sequence are less than A003418(p). (Let p_(n) be a prime for which the inequality is satisfied, and let p_(n+1) be the smallest prime > (p_(n))^2. No number smaller than A003418(p_(n+1)) can belong to this sequence. However, for any p_(n) that satisfies the inequality, so does p_(n+1), leading to an endless cycle.) This inequality is first satisfied at p=53, as A045948(53)=5040 > 53^2=2809.
Proof: It follows from the definitions of p_(n) and p_(n+1), and from Bertrand's Postulate, that 2(A045948(p_(n))) > 2((p_(n))^2) > p_(n+1). Therefore 2((A045948(p_(n)))^2 > (p_(n+1))^2.
Since any prime that divides A003418(p_(n)) divides A003418(p_(n+1)) at least twice as often, A045948(p_(n+1)) cannot be less than the product of (A045948(p_n))^2 and A034386(p_(n)). (The latter term greatly exceeds 2 for any actual p_(n).)
Therefore A045948(p_(n+1)) > 2((A045948(p_n))^2 > (p_(n+1))^2, and p_(n+1) satisfies the inequality, implying that no number smaller than A003418(p_(n+2)) can belong to this sequence.

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Also intersection of A003418 and A060735, and of A003418 and A168264. (A168264 is a subsequence of A060735, which is a subsequence of A116998.)

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 72, 72

Offset: 1

Gus Wiseman, Jan 09 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(8) = 3 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,7,8}.

Ana Rechtman, Décembre 2020, 4e défi (in French), Images des Mathématiques, CNRS, 2020.

Rightmost terms of A186974 and A320436.
Run lengths are A053707.
Cf. A015614, A036234, A051424, A085945, A186971, A186972, A186994, A276187, A303139, A320423, A320426.
Cf. A000005, A045948.

Table[Length[Select[Subsets[Range[n],{PrimePi[n]+1}],CoprimeQ@@#&]],{n,24}] (* see A186974 for a faster program *)

a(n) = prod(p=1, n, if (isprime(p), logint(n, p), 1)); \\ Michel Marcus, Dec 26 2020

a(n) = Product_{p prime <= n} floor(log_p(n)).

a(n) = A000005(A045948(n)). - Ridouane Oudra, Sep 02 2019

cp's OEIS Frontend

A048148 Distinct elements of A045948.

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A049614 n! divided by its squarefree kernel.

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A056813 Largest non-unitary prime factor of LCM(1,...,n); that is, the largest prime which occurs to power > 1 in prime factorization of LCM(1,..,n).

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A168262 Intersection of A003418 and A116998.

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A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

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