A102631 a(n) = n^2 / (squarefree kernel of n).

1, 2, 3, 8, 5, 6, 7, 32, 27, 10, 11, 24, 13, 14, 15, 128, 17, 54, 19, 40, 21, 22, 23, 96, 125, 26, 243, 56, 29, 30, 31, 512, 33, 34, 35, 216, 37, 38, 39, 160, 41, 42, 43, 88, 135, 46, 47, 384, 343, 250, 51, 104, 53, 486, 55, 224, 57, 58, 59, 120, 61, 62, 189, 2048, 65, 66, 67

Offset: 1

Author

Reinhard Zumkeller, Feb 25 2005

Keywords

• nonn
• easy
• mult

Comments

Index of first occurrence of n in A019554. - Franklin T. Adams-Watters, Nov 17 2006

Links

• Michel Marcus, Table of n, a(n) for n = 1..5000

Crossrefs

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

Programs

Mathematica

a[n_] := n^2/Times @@ FactorInteger[n][[All, 1]];
Array[a, 70] (* Jean-François Alcover, Jun 11 2019 *)

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

Sage

def A102631(n) :
    p = n
    for a in factor(n) :
        if a[1] > 1 :
            p = p * a[0]^(a[1]-1)
    return p
[A102631(n) for n in (1..67)] # Peter Luschny, Feb 07 2012

Formula

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

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

Multiplicative with a(p^e) = p^{2e-1}. - Franklin T. Adams-Watters, Nov 17 2006

Dirichlet g.f.: Product_{p prime} (1 - p/(p^2 - p^s)). - Amiram Eldar, Aug 28 2023

a(n) = A350390(n^2). - Amiram Eldar, Nov 30 2023