cp's OEIS Frontend

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

%I A007947 #281 Aug 02 2025 10:03:22
%S A007947 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,
%T A007947 29,30,31,2,33,34,35,6,37,38,39,10,41,42,43,22,15,46,47,6,7,10,51,26,
%U A007947 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
%N A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.
%C A007947 Multiplicative with a(p^e) = p.
%C A007947 Product of the distinct prime factors of n.
%C A007947 a(k)=k for k=squarefree numbers A005117. - _Lekraj Beedassy_, Sep 05 2006
%C A007947 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]
%C A007947 a(n) = A128651(A129132(n-1) + 2) for n > 1. - _Reinhard Zumkeller_, Mar 30 2007
%C A007947 Also the least common multiple of the prime factors of n. - _Peter Luschny_, Mar 22 2011
%C A007947 The Mobius transform of the sequence generates the sequence of absolute values of A097945. - _R. J. Mathar_, Apr 04 2011
%C A007947 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
%C A007947 a(n) differs from A014963(n) when n is a term of A024619. - _Eric Desbiaux_, Mar 24 2014
%C A007947 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
%C A007947 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
%H A007947 Daniel Forgues, <a href="/A007947/b007947.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from T. D. Noe)
%H A007947 Masum Billal, <a href="http://arxiv.org/abs/1501.00609">Divisible Sequence and its Characteristic Sequence</a>, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
%H A007947 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>
%H A007947 Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]
%H A007947 Jarosław Grytczuk, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.039">Thue type problems for graphs, points and numbers</a>, Discrete Math., 308 (2008), 4419-4429.
%H A007947 Neville Holmes, <a href="http://www.comp.utas.edu.au/users/nholmes/sqncs/index.htm#A007947">Integer Sequences</a> [Broken link]
%H A007947 Serge Lang, <a href="http://dx.doi.org/10.1090/S0273-0979-1990-15899-9">Old and New Conjectured Diophantine Inequalities</a>, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
%H A007947 Wolfdieter Lang, <a href="https://arxiv.org/abs/2307.10645">Cantor's List of Real Algebraic Numbers of Heights 1 to 7</a>, arXiv:2307.10645 [math.NT], 2023.
%H A007947 D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf">Euler constants for arithmetical progressions</a>, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
%H A007947 Ivar Peterson, <a href="http://web.archive.org/web/20130702061828/http://www.maa.org/mathland/mathtrek_12_8.html">The Amazing ABC Conjecture</a>
%H A007947 Paul Tarau, <a href="http://dx.doi.org/10.1007/978-3-642-23283-1_15">Emulating Primality with Multiset Representations of Natural Numbers</a>, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
%H A007947 Paul Tarau, <a href="http://dx.doi.org/10.1016/j.tcs.2014.04.025">Towards a generic view of primality through multiset decompositions of natural numbers</a>, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
%H A007947 Wikipedia, <a href="https://en.wikipedia.org/wiki/Radical_of_an_integer">Radical of an integer</a>.
%F A007947 If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
%F A007947 a(n) = Product_{k=1..A001221(n)} A027748(n,k). - _Reinhard Zumkeller_, Aug 27 2011
%F A007947 Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - _R. J. Mathar_, Jan 21 2012
%F A007947 a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - _Enrique Pérez Herrero_, Apr 23 2012
%F A007947 a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - _Michel Marcus_, Jan 06 2015
%F A007947 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
%F A007947 a(n) = n/A003557(n). - _Juri-Stepan Gerasimov_, Apr 07 2017
%F A007947 G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Apr 11 2017
%F A007947 From _Antti Karttunen_, Jun 18 2017: (Start)
%F A007947 a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
%F A007947 a(n) = A019565(A087207(n)). (End)
%F A007947 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
%F A007947 From _Peter Munn_, Jan 01 2020: (Start)
%F A007947 a(A059896(n,k)) = A059896(a(n), a(k)).
%F A007947 a(A003961(n)) = A003961(a(n)).
%F A007947 a(n^2) = a(n).
%F A007947 a(A225546(n)) = A019565(A267116(n)). (End)
%F A007947 Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - _Vaclav Kotesovec_, Jun 24 2020
%F A007947 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A007947 a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
%F A007947 a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
%F A007947 For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
%F A007947 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)
%F A007947 a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - _Ridouane Oudra_, Aug 01 2025
%e A007947 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
%p A007947 with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
%p A007947 A007947 := n -> ilcm(op(numtheory[factorset](n))):
%p A007947 seq(A007947(i),i=1..69); # _Peter Luschny_, Mar 22 2011
%p A007947 A:= n -> convert(numtheory:-factorset(n),`*`):
%p A007947 seq(A(n),n=1..100); # _Robert Israel_, Aug 10 2014
%p A007947 seq(NumberTheory:-Radical(n), n = 1..78); # _Peter Luschny_, Jul 20 2021
%t A007947 rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* _Robert G. Wilson v_, Aug 29 2012 *)
%t A007947 Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* _Harvey P. Dale_, Jul 14 2014 *)
%t A007947 a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* _Michael Somos_, Jul 15 2018 *)
%t A007947 Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* _Vaclav Kotesovec_, May 20 2020 *)
%o A007947 (PARI) a(n) = factorback(factorint(n)[,1]); \\ _Andrew Lelechenko_, May 09 2014
%o A007947 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020
%o A007947 (Magma) [ &*PrimeDivisors(n): n in [1..100] ]; // _Klaus Brockhaus_, Dec 04 2008
%o A007947 (Haskell)
%o A007947 a007947 = product . a027748_row  -- _Reinhard Zumkeller_, Feb 27 2012
%o A007947 (Sage) def A007947(n): return mul(p for p in prime_divisors(n))
%o A007947 [A007947(n) for n in (1..60)] # _Peter Luschny_, Mar 07 2017
%o A007947 (Python)
%o A007947 from sympy import primefactors, prod
%o A007947 def a(n): return 1 if n < 2 else prod(primefactors(n))
%o A007947 [a(n) for n in range(1, 51)]  # _Indranil Ghosh_, Apr 16 2017
%o A007947 (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
%Y A007947 See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
%Y A007947 More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
%Y A007947 Range of values is A005117.
%Y A007947 Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
%Y A007947 Bisections: A099984, A099985.
%Y A007947 Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
%Y A007947 A003961, A059896 are used to express relationship between terms of this sequence.
%Y A007947 Cf. A001615, A008683, A076479, A078615.
%K A007947 nonn,easy,nice,mult
%O A007947 1,2
%A A007947 R. Muller, Mar 15 1996
%E A007947 More terms from several people including _David W. Wilson_
%E A007947 Definition expanded by _Jonathan Sondow_, Apr 26 2013