A342424 -id:A342424 - cp's OEIS frontend

A342423 a(n) = Sum_{k=1..n} gcd(k,n)^gcd(k,n).

1, 5, 29, 262, 3129, 46693, 823549, 16777484, 387420549, 10000003145, 285311670621, 8916100495490, 302875106592265, 11112006826381589, 437893890380865741, 18446744073726329368, 827240261886336764193, 39346408075296925089309

Offset: 1

Seiichi Manyama, Mar 11 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A000010, A332517, A342420, A342424.

a[n_] := Sum[GCD[k, n]^GCD[k, n], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Mar 11 2021 *)

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

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

a(n) = Sum_{d|n} phi(n/d) * d^d.

A342437 a(n) = Sum_{k=1..n} gcd(k,n)^(n/gcd(k,n) - 1).

Original entry on oeis.org

1, 2, 3, 5, 5, 14, 7, 25, 25, 74, 11, 161, 13, 398, 383, 657, 17, 2110, 19, 3341, 4485, 10262, 23, 19569, 2521, 49178, 39547, 74441, 29, 221462, 31, 328737, 590753, 1048610, 103379, 1905565, 37, 4718630, 6377655, 5573801, 41, 22462826, 43, 31459985, 40634221, 92274734, 47

Offset: 1

Seiichi Manyama, Mar 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A000010, A226459, A342421, A342424.

a[n_] := Sum[GCD[k, n]^(n/GCD[k, n] - 1), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Mar 12 2021 *)

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

a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n/d-1));

a(n) = Sum_{d|n} phi(n/d) * d^(n/d-1).

If p is prime, a(p) = p.

A344195 a(n) = Sum_{k=1..n} tau(gcd(k,n))^(n/gcd(k,n)), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 3, 4, 9, 6, 26, 8, 49, 25, 140, 12, 240, 14, 782, 156, 1215, 18, 3349, 20, 5130, 800, 20498, 24, 19558, 151, 98324, 3148, 111492, 30, 270624, 32, 551091, 20520, 2097176, 924, 1716189, 38, 9437210, 98348, 8630496, 42, 25362724, 44, 43714584, 266346, 184549406, 48, 137141048, 813, 671096867

Offset: 1

Seiichi Manyama, May 11 2021

nonn

Cf. A000005, A279789, A342424, A344140, A344194.

a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^(n/#) &]; Array[a, 50] (* Amiram Eldar, May 11 2021 *)

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

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

a(n) = Sum_{d|n} phi(n/d) * tau(d)^(n/d).

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

cp's OEIS Frontend

A342423 a(n) = Sum_{k=1..n} gcd(k,n)^gcd(k,n).

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A342437 a(n) = Sum_{k=1..n} gcd(k,n)^(n/gcd(k,n) - 1).

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Mathematica

PARI

PARI

Formula

A344195 a(n) = Sum_{k=1..n} tau(gcd(k,n))^(n/gcd(k,n)), where tau(n) is the number of divisors of n.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula