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A067240 If n = Product_{i} p_i^e_i, a(n) = Sum_{i} (p_i - 1)*p_i^(e_i - 1). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Additive sequence with a(p^e) = phi(p^e) = (p-1)*p^(e-1). - Charles R Greathouse IV, Sep 10 2015
LINKS
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186. (T on p. 181.)
FORMULA
For n > 1: a(n) = Sum_{i} phi(p_i^e_i). - T. D. Noe, Jul 10 2003
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 *)
PROG
(PARI)
{
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
(Haskell)
a067240 1 = 0
a067240 n = sum $ map a000010 $ a141809_row $ toInteger n
-- Reinhard Zumkeller, Jun 13 2012
(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
CROSSREFS
Sequence in context: A334952 A220096 A122376 * A126080 A302043 A060681
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 10 2002
STATUS
approved

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Last modified July 24 10:37 EDT 2024. Contains 374583 sequences. (Running on oeis4.)