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A342421 a(n) = Sum_{k=1..n} (n/gcd(k,n))^(gcd(k,n) - 1).

%I A342421 #19 Mar 12 2021 09:48:38
%S A342421 1,2,3,5,5,13,7,21,25,41,11,135,13,113,271,297,17,875,19,1573,1765,
%T A342421 1145,23,9215,2521,4265,13627,18539,29,71371,31,67729,119329,65825,
%U A342421 76931,637061,37,262505,1064935,1381637,41,4432817,43,4207855,11169629,4194833,47
%N A342421 a(n) = Sum_{k=1..n} (n/gcd(k,n))^(gcd(k,n) - 1).
%H A342421 Seiichi Manyama, <a href="/A342421/b342421.txt">Table of n, a(n) for n = 1..5000</a>
%F A342421 a(n) = Sum_{d|n} phi(d^(n/d)) = Sum_{d|n} phi(d) * d^(n/d-1).
%F A342421 If p is prime, a(p) = p.
%t A342421 a[n_] := Sum[(n/GCD[k, n])^(GCD[k, n] - 1), {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Mar 11 2021 *)
%o A342421 (PARI) a(n) = sum(k=1, n, (n/gcd(k, n))^(gcd(k, n)-1));
%o A342421 (PARI) a(n) = sumdiv(n, d, eulerphi(d^(n/d)));
%o A342421 (PARI) a(n) = sumdiv(n, d, eulerphi(d)*d^(n/d-1));
%Y A342421 Cf. A000010, A342422.
%K A342421 nonn
%O A342421 1,2
%A A342421 _Seiichi Manyama_, Mar 11 2021