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
Examples
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 + ...
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]
Programs
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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
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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 *)
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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 */
Formula
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
Extensions
More terms from Enrique Pérez Herrero, Apr 03 2014
Comments