A067240 If n = Product_{i} p_i^e_i, a(n) = Sum_{i} (p_i - 1)*p_i^(e_i - 1).

0, 1, 2, 2, 4, 3, 6, 4, 6, 5, 10, 4, 12, 7, 6, 8, 16, 7, 18, 6, 8, 11, 22, 6, 20, 13, 18, 8, 28, 7, 30, 16, 12, 17, 10, 8, 36, 19, 14, 8, 40, 9, 42, 12, 10, 23, 46, 10, 42, 21, 18, 14, 52, 19, 14, 10, 20, 29, 58, 8, 60, 31, 12, 32, 16, 13, 66, 18, 24, 11, 70, 10, 72, 37, 22, 20, 16, 15, 78, 12, 54, 41, 82, 10, 20, 43, 30, 14, 88, 11, 18, 24, 32, 47, 22, 18, 96, 43, 16, 22

Offset: 1

Author

N. J. A. Sloane, Mar 10 2002

Keywords

• nonn

Comments

Additive sequence with a(p^e) = phi(p^e) = (p-1)*p^(e-1). - Charles R Greathouse IV, Sep 10 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186. (T on p. 181.)

Crossrefs

Cf. A000010, A141809.

Programs

Haskell

a067240 1 = 0
a067240 n = sum $ map a000010 $ a141809_row $ toInteger n
-- Reinhard Zumkeller, Jun 13 2012

Maple

with(numtheory); A067240 := proc(n) local e,j; e := ifactors(n)[2]: add((e[j][1]-1)*e[j][1]^(e[j][2]-1),j=1..nops(e)); end;

Mathematica

a[n_] := Total[ EulerPhi[ Power @@ #] & /@ FactorInteger[n]]; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 22 2012, after T. D. Noe *)

PARI

A067240(n)=
{
    local(f=factor(n), r=0, p, e);
    for (i=1, matsize(f)[1],
        p=f[i,1];  e=f[i,2];
        r += (p-1)*p^(e-1);
    );
    return(r);
} /* Joerg Arndt, Jun 10 2011 */

PARI

a(n)=my(f=factor(n)); sum(i=1,#f~, (f[i,1]-1)*f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Sep 10 2015

Python

from sympy import factorint
a = lambda n: sum((p**(e-1))*(p-1) for p,e in factorint(n).items() if e > 0) # Darío Clavijo, Feb 15 2024

Formula

For n > 1: a(n) = Sum_{i} phi(p_i^e_i). - T. D. Noe, Jul 10 2003