|
|
A007947
|
|
Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.
|
|
961
|
|
|
1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), lcm(b,c) = A007947(n) = "squarefree kernel" of n and b*c = A019554(n) = "outer square root" of n.
The characterization a(n) = lcm(b,c) from the above is incorrect. It fails when n is biquadrateful (A046101). As an example, when n = 48 then sqrt(48) = 4*sqrt(3), so b = 4 and c = 3; however a(48) = 6 is not lcm(4, 3). - Jeppe Stig Nielsen, Oct 10 2021
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length. For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017
|
|
LINKS
|
|
|
FORMULA
|
If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
a(n^2) = a(n).
(End)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
|
|
EXAMPLE
|
G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - Michael Somos, Jul 15 2018
|
|
MAPLE
|
with(numtheory); A007947 := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1], i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
A:= n -> convert(numtheory:-factorset(n), `*`):
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021
|
|
MATHEMATICA
|
rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n], SquareFreeQ]], {n, 100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)
|
|
PROG
|
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020
(Magma) [ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008
(Haskell)
(Sage) def A007947(n): return mul(p for p in prime_divisors(n))
(Python)
from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
|
|
CROSSREFS
|
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
A003961, A059896 are used to express relationship between terms of this sequence.
|
|
KEYWORD
|
nonn,easy,nice,mult
|
|
AUTHOR
|
R. Muller, Mar 15 1996
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|