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a(n) = Sum_{k=1..n} k^(gcd(k,n) - 1).
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%I #18 Mar 10 2021 12:28:44

%S 1,3,11,68,629,7793,117655,2097228,43046772,1000000649,25937424611,

%T 743008379146,23298085122493,793714773371841,29192926025401528,

%U 1152921504608945960,48661191875666868497,2185911559738739835591,104127350297911241532859

%N a(n) = Sum_{k=1..n} k^(gcd(k,n) - 1).

%F If p is prime, a(p) = p-1 + p^(p-1) = A173235(p) = A056665(p).

%t a[n_] := Sum[k^(GCD[k, n] - 1), {k, 1, n}]; Array[a, 19] (* _Amiram Eldar_, Mar 10 2021 *)

%o (PARI) a(n) = sum(k=1, n, k^(gcd(k, n)-1));

%Y Cf. A056665, A173235, A342389, A342396.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Mar 10 2021