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%I #47 Apr 03 2022 13:44:51
%S 1,4,6,11,10,24,14,28,26,40,22,66,26,56,60,68,34,104,38,110,84,88,46,
%T 168,74,104,102,154,58,240,62,160,132,136,140,286,74,152,156,280,82,
%U 336,86,242,260,184,94,408,146,296,204,286,106,408,220,392,228,232,118,660
%N a(n) = Sum_{d|n} psi(n/d) * phi(d).
%C Dirichlet convolution of Dedekind psi function (A001615) with Euler totient function (A000010).
%C Dirichlet convolution of A008966 with A018804.
%C Dirichlet convolution of A038040 with A271102.
%H Robert Israel, <a href="/A322577/b322577.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindFunction.html">Dedekind Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%F Dirichlet g.f.: zeta(s-1)^2 / zeta(2*s).
%F a(p) = 2*p, where p is prime.
%F Sum_{k=1..n} a(k) ~ 45*n^2*(2*Pi^4*log(n) - Pi^4 + 4*gamma*Pi^4 - 360*zeta'(4)) / (2*Pi^8), where gamma is the Euler-Mascheroni constant A001620 and for zeta'(4) see A261506. - _Vaclav Kotesovec_, Aug 31 2019
%F a(p^k) = (k+1)*p^k - (k-1)*p^(k-2) where p is prime. - _Robert Israel_, Sep 01 2019
%F a(n) = Sum_{k=1..n} psi(gcd(n,k)). - _Ridouane Oudra_, Nov 29 2019
%F a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 07 2021
%p f:= proc(n) local t;
%p mul((t[2]+1)*t[1]^t[2] - (t[2]-1)*t[1]^(t[2]-2), t = ifactors(n)[2])
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Sep 01 2019
%t Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 60}]
%t f[p_, e_] := (e + 1)*p^e - (e - 1)*p^(e - 2); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 26 2020 *)
%o (PARI) seq(n) = {dirmul(vector(n, n, eulerphi(n)), vector(n, n, n * sumdivmult(n, d, issquarefree(d)/d)))} \\ _Andrew Howroyd_, Aug 29 2019
%o (PARI)
%o A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
%o A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d)); \\ _Antti Karttunen_, Apr 03 2022
%Y Cf. A000010, A001615, A008966, A018804, A038040, A271102.
%Y Cf. A327251 (inverse Möbius transform), A347092 (Dirichlet inverse), A347093 (sum with it), A347135.
%K nonn,mult,easy
%O 1,2
%A _Ilya Gutkovskiy_, Aug 29 2019