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

1,2

Comments

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

Links

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

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)

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A375931 A373231 A334684 * A327526 A378665 A121758

Adjacent sequences: A062756 A062757 A062758 * A062760 A062761 A062762

Keyword

nonn

Author

Labos Elemer, Jul 16 2001

Status

approved