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A130107
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Möbius transform of A063659.
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1
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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
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OFFSET
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1,3
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COMMENTS
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Double inverse Möbius transform of A130107 = A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, ...).
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LINKS
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FORMULA
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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
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EXAMPLE
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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 + ...
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MAPLE
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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:
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MATHEMATICA
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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[#]&];
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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PROG
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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 */
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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