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A349171 a(n) = Sum_{d|n} phi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and phi is Euler totient function.

%I A349171 #13 Nov 15 2021 13:08:08
%S A349171 1,4,6,14,10,24,14,46,30,40,22,84,26,56,60,146,34,120,38,140,84,88,46,
%T A349171 276,80,104,138,196,58,240,62,454,132,136,140,420,74,152,156,460,82,
%U A349171 336,86,308,300,184,94,876,154,320,204,364,106,552,220,644,228,232,118,840,122,248,420,1394,260,528,134,476,276
%N A349171 a(n) = Sum_{d|n} phi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and phi is Euler totient function.
%C A349171 Dirichlet convolution of A003959 with Euler totient function phi, A000010.
%C A349171 Möbius transform of A349170.
%H A349171 Antti Karttunen, <a href="/A349171/b349171.txt">Table of n, a(n) for n = 1..20000</a>
%F A349171 a(n) = Sum_{d|n} A000010(d) * A003959(n/d).
%F A349171 a(n) = Sum_{d|n} A008683(d) * A349170(n/d).
%F A349171 a(n) = Sum_{k=1..n} A003959(gcd(n, k)).
%F A349171 a(n) = A018804(n) + A349141(n).
%F A349171 For all n >= 1, a(n) >= A349131(n).
%F A349171 Multiplicative with a(p^e) = p*(p+1)^e - (p-1)*p^e. - _Amiram Eldar_, Nov 09 2021
%t A349171 f[p_, e_] := p*(p + 1)^e - (p - 1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2021 *)
%o A349171 (PARI)
%o A349171 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A349171 A349171(n) = sumdiv(n,d,eulerphi(d)*A003959(n/d));
%Y A349171 Cf. A000010, A003959, A018804, A349141, A349170 (inverse Möbius transform), A349172, A349131.
%K A349171 nonn,mult
%O A349171 1,2
%A A349171 _Antti Karttunen_, Nov 09 2021