login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k). 100
1, 1, 2, 1, 4, 2, 6, 1, 4, 4, 10, 2, 12, 6, 8, 1, 16, 4, 18, 4, 12, 10, 22, 2, 16, 12, 8, 6, 28, 8, 30, 1, 20, 16, 24, 4, 36, 18, 24, 4, 40, 12, 42, 10, 16, 22, 46, 2, 36, 16, 32, 12, 52, 8, 40, 6, 36, 28, 58, 8, 60, 30, 24, 1, 48, 20, 66, 16, 44, 24, 70, 4, 72, 36, 32, 18, 60, 24, 78, 4, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Completely multiplicative.

Dirichlet inverse of A097945. - R. J. Mathar, Aug 29 2011

LINKS

T. D. Noe and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)

Vaclav Kotesovec, Graph - the asymptotic ratio (10^7 terms)

Index to divisibility sequences

FORMULA

Multiplicative with a(p^e) = (p-1)^e. - David W. Wilson, Aug 01 2001

a(n) = A000010(n) iff n is squarefree (see A005117). - Reinhard Zumkeller, Nov 05 2004

a(n) = abs(A125131(n)). - Tom Edgar, May 26 2014

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4 / (315 * zeta(3)) = 0.25725505075419... - Vaclav Kotesovec, Jun 14 2020

Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(1-s) + p^(-s)). - Ilya Gutkovskiy, Feb 27 2022

MAPLE

a:= n-> mul((i[1]-1)^i[2], i=ifactors(n)[2]):

seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

MATHEMATICA

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[ f[n/d]*DirichletInverse[f][d], {d, Most[ Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; Table[ DirichletInverse[ muphi][n], {n, 1, 81}] (* Jean-Fran├žois Alcover, Dec 12 2011, after R. J. Mathar *)

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 10 2016 *)

PROG

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-p*X+X))[n]) /* Ralf Stephan */

(Haskell)

a003958 1 = 1

a003958 n = product $ map (subtract 1) $ a027746_row n

-- Reinhard Zumkeller, Apr 09 2012, Mar 02 2012

(Python)

from math import prod

from sympy import factorint

def a(n): return prod((p-1)**e for p, e in factorint(n).items())

print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Feb 27 2022

CROSSREFS

Cf. A003959, A168065, A168066, A027746, A006093, A027748, A124010.

Sequence in context: A187202 A345046 A125131 * A326140 A082729 A326069

Adjacent sequences:  A003955 A003956 A003957 * A003959 A003960 A003961

KEYWORD

nonn,mult,nice

AUTHOR

Marc LeBrun

EXTENSIONS

Definition reedited (from formula) by Daniel Forgues, Nov 17 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 3 17:31 EDT 2022. Contains 355055 sequences. (Running on oeis4.)