|
%I #21 Feb 12 2023 03:15:56
%S 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,
%T 14,29,30,31,32,33,34,35,36,37,38,39,10,41,42,43,22,15,46,47,6,49,10,
%U 51,26,53,6,55,14,57,58,59,30,61,62,21,64,65,66,67,34,69,70,71,36,73
%N Largest power of squarefree kernel of n (= A007947) which divides n.
%C a(n) is a first power if and only if n is not a powerful number (A001694, A052485).
%H Reinhard Zumkeller, <a href="/A062759/b062759.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A007947(n)^A051904(n).
%F From _Amiram Eldar_, Feb 12 2023: (Start)
%F a(n) = n/A062759(n).
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 / 2 = 0.352221... . (End)
%e 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.
%t {1}~Join~Table[#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 73}] (* _Michael De Vlieger_, Nov 02 2017 *)
%t 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 *)
%o (Haskell)
%o a062759 n = a007947 n ^ a051904 n -- _Reinhard Zumkeller_, Jul 15 2012
%o (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
%Y Cf. A001694, A003557, A007947, A051904, A052485, A062759, A065463.
%K nonn
%O 1,2
%A _Labos Elemer_, Jul 16 2001
|
