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A130107 Möbius transform of A063659. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A225368 A099311 A130742 * A107130 A194747 A065423

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

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Last modified February 24 17:24 EST 2020. Contains 332209 sequences. (Running on oeis4.)