A130107 Möbius transform of A063659.

1, 1, 2, 1, 4, 2, 6, 3, 5, 4, 10, 2, 12, 6, 8, 6, 16, 5, 18, 4, 12, 10, 22, 6, 19, 12, 16, 6, 28, 8, 30, 12, 20, 16, 24, 5, 36, 18, 24, 12, 40, 12, 42, 10, 20, 22, 46, 12, 41, 19, 32, 12, 52, 16, 40, 18, 36, 28, 58, 8, 60, 30, 30, 24, 48, 20, 66, 16, 44, 24

Offset

1,3

Comments

Double inverse Möbius transform of A130107 = A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, ...).

Links

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000

Steven R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]

Formula

A054525 * A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...).

Multiplicative with a(p^e) = p-1 if e=1, a(p^e) = p^2-p-1 if e=2, a(p^e) = p^(e-3)*(p+1)*(p-1)^2. - Enrique Pérez Herrero, Apr 03 2014

Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(2s)). - Álvar Ibeas, Mar 07 2015

Sum_{k=1..n} a(k) ~ 270*n^2 / Pi^6. - Vaclav Kotesovec, Jan 11 2019

Example

G.f. = x + x^2 + 2*x^3 + x^4 + 4*x^5 + 2*x^6 + 6*x^7 + 3*x^8 + 5*x^9 + ...

Maple

with(numtheory): A130107 := proc(n) local dp, mtdp, d, p;
dp := n -> n*mul((1+1/p), p=factorset(n));
mtdp := n -> add(mobius(n/d)*dp(d), d=divisors(n));
add(mobius(n/d)*mtdp(d), d=divisors(n)) end:
seq(A130107(n), n=1..76); # Peter Luschny, Apr 06 2014

Mathematica

JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n];
DedekindPsi[n_]:=JordanTotient[n, 2]/EulerPhi[n];
A063659[n_]:=DivisorSum[n, MoebiusMu[n/#]*DedekindPsi[#]&];
A130107[n_]:=DivisorSum[n, MoebiusMu[n/#]*A063659[#]&]:
Table[A130107[n], {n, 1, 30}]
(* Enrique Pérez Herrero, Apr 03 2014 *)
a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ #2 == 1, # - 1, #2 == 2, #^2 - # - 1, True, #^(#2 - 3) (#^2 - 1) (# - 1)] &) @@@ FactorInteger[n]]; (* Michael Somos, Jun 17 2015 *)

Prog

(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( e==1, p - 1, e==2, p^2 - p - 1, p^(e-3) * (p^2 - 1) * (p-1))))}; /* Michael Somos, Jun 17 2015 */

Crossrefs

Cf. A130107, A054525, A063659, A001615.

Sequence in context: A099311 A349442 A130742 * A375480 A107130 A194747

Adjacent sequences: A130104 A130105 A130106 * A130108 A130109 A130110

Keyword

nonn,mult

Author

Gary W. Adamson, May 07 2007

Extensions

More terms from Enrique Pérez Herrero, Apr 03 2014

Status

approved