A062759 Largest power of squarefree kernel of n (= A007947) which divides n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 25, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 49, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69, 70, 71, 36, 73

Offset: 1

Author

Labos Elemer, Jul 16 2001

Keywords

• nonn

Comments

a(n) is a first power if and only if n is not a powerful number (A001694, A052485).

Examples

			n = 1800: squarefree kernel is 2*3*5 = 30 and a(1800) = 900 = 30^2 divides n, exponent of 30 is the smallest prime exponent of 1800 = 2*2*2*3*3*5*5.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A003557, A007947, A051904, A052485, A062759, A065463.

Programs

Haskell

a062759 n = a007947 n ^ a051904 n  -- Reinhard Zumkeller, Jul 15 2012

Mathematica

{1}~Join~Table[#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 73}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] = e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

PARI

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmin(f[,2])); prod(i = 1, #f~, f[i,1]^e));} \\ Amiram Eldar, Feb 12 2023

Formula

a(n) = A007947(n)^A051904(n).

From Amiram Eldar, Feb 12 2023: (Start)

a(n) = n/A062759(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 / 2 = 0.352221... . (End)