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%I #36 Aug 27 2023 02:51:10
%S 1,2,3,2,5,6,7,4,6,10,11,6,13,14,15,8,17,12,19,10,21,22,23,12,20,26,
%T 18,14,29,30,31,16,33,34,35,12,37,38,39,20,41,42,43,22,30,46,47,24,42,
%U 40,51,26,53,36,55,28,57,58,59,30,61,62,42,32,65,66,67,34,69,70
%N Möbius transform of A034448.
%H Álvar Ibeas, <a href="/A254503/b254503.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = phi(A057521(n)) * A055231(n).
%F If n is squarefree, a(n) = n; if n is powerful, a(n) = phi(n).
%F Multiplicative with a(p) = p; a(p^e) = phi(p^e), for e > 1.
%F Dirichlet g.f.: zeta(s-1) / zeta(2s-1).
%F a(n) = Sum_{d|n, gcd(n/d, d) = 1} mu(d)^2 * phi(n/d). - _Daniel Suteu_, Jun 27 2018
%F Sum_{k=1..n} a(k) ~ n^2 / (2*zeta(3)). - _Vaclav Kotesovec_, Jan 11 2019
%t Table[DivisorSum[n, MoebiusMu[#]^2*EulerPhi[n/#] &, CoprimeQ[n/#, #] &], {n, 70}] (* _Michael De Vlieger_, Jun 27 2018 *)
%t f[p_, e_] := (p - 1)*p^(e - 1); f[p_, 1] := p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 27 2023 *)
%o (PARI) a(n) = {my(f = factor(n)); for (i=1, #f~, if ((e=f[i, 2]) > 1, f[i, 1] = eulerphi(f[i, 1]^e); f[i, 2] = 1);); factorback(f);} \\ _Michel Marcus_, Feb 06 2015
%o (PARI) a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d))); \\ _Daniel Suteu_, Jun 27 2018
%Y Cf. A000010 (totient), A001694 (powerful), A005117 (squarefree), A034448 (usigma), A057521 (powerful part), A055231 (unitary squarefree kernel).
%K mult,nonn,easy
%O 1,2
%A _Álvar Ibeas_, Jan 31 2015
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