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a(n) = n^2 / (squarefree kernel of n).
10

%I #26 Nov 30 2023 02:53:45

%S 1,2,3,8,5,6,7,32,27,10,11,24,13,14,15,128,17,54,19,40,21,22,23,96,

%T 125,26,243,56,29,30,31,512,33,34,35,216,37,38,39,160,41,42,43,88,135,

%U 46,47,384,343,250,51,104,53,486,55,224,57,58,59,120,61,62,189,2048,65,66,67

%N a(n) = n^2 / (squarefree kernel of n).

%C Index of first occurrence of n in A019554. - _Franklin T. Adams-Watters_, Nov 17 2006

%H Michel Marcus, <a href="/A102631/b102631.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = A000290(n)/A007947(n) = n*A003557(n);

%F a(n) = n iff n is squarefree: a(A005117(n)) = A005117(n).

%F Multiplicative with a(p^e) = p^{2e-1}. - _Franklin T. Adams-Watters_, Nov 17 2006

%F Dirichlet g.f.: Product_{p prime} (1 - p/(p^2 - p^s)). - _Amiram Eldar_, Aug 28 2023

%F a(n) = A350390(n^2). - _Amiram Eldar_, Nov 30 2023

%t a[n_] := n^2/Times @@ FactorInteger[n][[All, 1]];

%t Array[a, 70] (* _Jean-François Alcover_, Jun 11 2019 *)

%o (Sage)

%o def A102631(n) :

%o p = n

%o for a in factor(n) :

%o if a[1] > 1 :

%o p = p * a[0]^(a[1]-1)

%o return p

%o [A102631(n) for n in (1..67)] # _Peter Luschny_, Feb 07 2012

%o (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,2] = 2*f[k,2]-1); factorback(f); \\ _Michel Marcus_, Aug 20 2017

%Y Cf. A000290, A007947, A003557, A005117, A019554, A350390.

%K nonn,easy,mult

%O 1,2

%A _Reinhard Zumkeller_, Feb 25 2005