

A007947


Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.


351



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
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OFFSET

1,2


COMMENTS

Multiplicative with a(p^e) = p.
For n>1, product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117.  Lekraj Beedassy, Sep 05 2006
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 bc = A019554(n) = "outer square root" of n.
a(n) = A128651(A129132(n1) + 2) for n>1.  Reinhard Zumkeller, Mar 30 2007
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) differs from A014963(n) when n is a term of A024619.  Eric Desbiaux, Mar 24 2014
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


LINKS

T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandachetype multiplicative sequences
Steven R. Finch, Unitarism and infinitarism.
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 44194429.
Neville Holmes, Integer Sequences
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 3775. see p. 39.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105124.
Wikipedia, Radical of an integer.


FORMULA

n = Product (p_j^k_j) > Product (p_j).
a(n) = Product(A027748(n,k): 1 <= k <= A001221(n)).  Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*product_{primes p} (1+p^(1s)p^(s)).  R. J. Mathar, Jan 21 2012
a(n) = sum(dn, phi(d) * mu(d)^2).  Enrique Pérez Herrero, Apr 23 2012
a(n) = prod(dn, 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((n1)/k))*A010051(k)*M(k)] where M(n) is the Mangoldt function.  Anthony Browne, Jun 17 2016


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))):
seq(A007947(i), i=1..69); # Peter Luschny, Mar 22 2011
A:= n > convert(numtheory:factorset(n), `*`):
seq(A(n), n=1..100); # Robert Israel, Aug 10 2014


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 *)


PROG

(PARI) a(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
(MAGMA) [ &*PrimeDivisors(n): n in [1..100] ]; /* Klaus Brockhaus, Dec 04 2008 */
(Haskell)
a007947 = product . a027748_row  Reinhard Zumkeller, Feb 27 2012
(Sage) def A007947(n) : return 1/mul(1/p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Jun 10 2012


CROSSREFS

Cf. A048803, A007913, A062953, A000188, A019554, A020500, A053462, A010051.
Bisection: A099984, A099985.
Sequence in context: A056554 A088835 * A015053 A062953 A015052 A053166
Adjacent sequences: A007944 A007945 A007946 * A007948 A007949 A007950


KEYWORD

nonn,easy,nice,mult


AUTHOR

R. Muller, Mar 15 1996


EXTENSIONS

More terms from several people including David W. Wilson
Definition expanded by Jonathan Sondow, Apr 26 2013


STATUS

approved



