A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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

Offset: 1

Author

R. Muller, Mar 15 1996

Keywords

• nonn
• easy
• nice
• mult

Comments

Multiplicative with a(p^e) = p.

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), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]

a(n) = A128651(A129132(n-1) + 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

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

Examples

			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
		

Links

• 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 Smarandache-type multiplicative sequences
• Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
• Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
• Neville Holmes, Integer Sequences [Broken link]
• Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
• Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
• D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. 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, 218-238
• Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
• Wikipedia, Radical of an integer.

Crossrefs

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.

More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.

Range of values is A005117.

Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.

Bisections: A099984, A099985.

Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).

A003961, A059896 are used to express relationship between terms of this sequence.

Cf. A001615, A008683, A076479, A078615.

Programs

Haskell

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

Magma

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

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
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 *)

PARI

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Python

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

Sage

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

Scheme

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

Formula

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).

a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011

Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012

a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 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

a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017

G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017

From Antti Karttunen, Jun 18 2017: (Start)

a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).

a(n) = A019565(A087207(n)). (End)

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

From Peter Munn, Jan 01 2020: (Start)

a(A059896(n,k)) = A059896(a(n), a(k)).

a(A003961(n)) = A003961(a(n)).

a(n^2) = a(n).

a(A225546(n)) = A019565(A267116(n)). (End)

Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020

From Richard L. Ollerton, May 07 2021: (Start)

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)

a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

Extensions

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013