A003958 - OEIS

A003958

If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).

102

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

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OFFSET

1,3

COMMENTS

Completely multiplicative.

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

LINKS

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)) = 1/(2*A082695) = 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

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + (p^(1-s) - 2) / (1 - p + p^s)), (with a product that converges for s=2). - Vaclav Kotesovec, Feb 11 2023

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-François 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