A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).

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

Offset: 1

Author

Marc LeBrun

Keywords

• nonn
• mult
• nice

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

Crossrefs

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

Programs

Haskell

a003958 1 = 1
a003958 n = product $ map (subtract 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Mar 02 2012

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 *)

PARI

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

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

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

Extensions

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