A344080 -id:A344080 - cp's OEIS frontend

A344140 a(n) = Sum_{x_1|n, x_2|n, ... , x_n|n} gcd(x_1,x_2, ... ,x_n).

1, 5, 10, 99, 36, 4290, 134, 72613, 20713, 1053700, 2058, 2194638822, 8204, 268550150, 1073938440, 156969213515, 131088, 101697785139535, 524306, 3657271905119820, 4398063288332, 17592232181770, 8388630, 4727105990672866963914, 847422827191, 4503600499785740

Offset: 1

Seiichi Manyama, May 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..719

Cf. A000203, A060724, A344080, A344132, A344138, A344139.

a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^n &]; Array[a, 20] (* Amiram Eldar, May 10 2021 *)

a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^n);

a(n) = sum(k=1, n, numdiv(gcd(k, n))^n);

a(n) = Sum_{x_1|n, x_2|n, ... , x_n|n} n/lcm(x_1,x_2, ... ,x_n).
a(n) = Sum_{d|n} phi(n/d) * tau(d)^n.
If p is prime, a(p) = 2^p - 1 + p.
a(n) = Sum_{k=1..n} tau(gcd(k,n))^n.

A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 9, 86, 33, 4109, 129, 65622, 19692, 1048613, 2049, 2176786526, 8193, 268435589, 1073741865, 152587956247, 131073, 101559956692208, 524289, 3656158441111670, 4398046511241, 17592186046469, 8388609, 4722366482871822065758

Offset: 1

Seiichi Manyama, May 09 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..719

Cf. A007425, A062367, A097988, A279789, A344047, A344080.

a[n_] := DivisorSum[n, DivisorSigma[0, #]^# &]; Array[a, 24] (* Amiram Eldar, May 09 2021 *)

PARI
```
a(n) = sumdiv(n, d, numdiv(d)^d);
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (numdiv(k)*x)^k/(1-x^k)))

G.f.: Sum_{k >= 1} (tau(k) * x)^k/(1 - x^k).

If p is prime, a(p) = 1 + 2^p.

cp's OEIS Frontend

A344140 a(n) = Sum_{x_1|n, x_2|n, ... , x_n|n} gcd(x_1,x_2, ... ,x_n).

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A344081 a(n) = Sum_{d|n} tau(d)^d, where tau(n) is the number of divisors of n.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula