A344194 a(n) = Sum_{k=1..n} tau(gcd(k,n))^gcd(k,n), where tau(n) is the number of divisors of n.
1, 5, 10, 87, 36, 4114, 134, 65629, 19705, 1048628, 2058, 2176786622, 8204, 268435614, 1073741928, 152587956347, 131088, 101559956696337, 524306, 3656158441111964, 4398046511420, 17592186046514, 8388630, 4722366482871822135514, 847288609591, 4503599627378748
Offset: 1
Keywords
Programs
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Mathematica
a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^# &]; Array[a, 26] (* Amiram Eldar, May 11 2021 *)
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PARI
a(n) = sum(k=1, n, numdiv(gcd(k, n))^gcd(k, n));
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PARI
a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^d);
Formula
a(n) = Sum_{d|n} phi(n/d) * tau(d)^d.
If p is prime, a(p) = 2^p - 1 + p.