A342394 -id:A342394 - cp's OEIS frontend

A342370 a(n) = Sum_{k=1..n} gcd(k,n)^(k-1).

1, 3, 11, 68, 629, 7797, 117655, 2097254, 43046979, 1000000799, 25937424611, 743008402000, 23298085122493, 793714773374529, 29192926027528343, 1152921504613147242, 48661191875666868497, 2185911559739107208115, 104127350297911241532859, 5242880000000008181608132

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Martin Ehrenstein, Table of n, a(n) for n = 1..400

Cf. A000010, A018804, A056665, A173235, A341036, A342394, A342433, A342436, A342449.

a[n_] := Sum[GCD[k, n]^(k - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

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

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

a(19) and beyond from Martin Ehrenstein, Mar 13 2021

A342389 a(n) = Sum_{k=1..n} k^gcd(k,n).

Original entry on oeis.org

1, 5, 30, 264, 3135, 46709, 823564, 16777528, 387420759, 10000003265, 285311670666, 8916100500148, 302875106592331, 11112006826381965, 437893890380965260, 18446744073726350224, 827240261886336764313, 39346408075296928032645

Offset: 1

Seiichi Manyama, Mar 10 2021

Cf. A000217, A228640, A342394, A342395.

a[n_] := Sum[k^GCD[k, n], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Mar 10 2021 *)

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

If p is prime, a(p) = A000217(p-1) + p^p = (p-1)*p/2 + p^p.

A342396 a(n) = Sum_{k=1..n} k^(n/gcd(k,n) - 1).

Original entry on oeis.org

1, 2, 6, 31, 355, 3150, 67172, 904085, 22998481, 427799450, 14914341926, 287337926355, 13421957361111, 339940911160914, 15434209582905140, 493467700905592777, 28101527071305611529, 836396358233559195382

Offset: 1

Seiichi Manyama, Mar 10 2021

Cf. A031971, A332621, A342394, A342395.

a[n_] := Sum[k^(n/GCD[k, n] - 1), {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Mar 10 2021 *)

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

If p is prime, a(p) = A031971(p-1) + 1.

cp's OEIS Frontend

A342370 a(n) = Sum_{k=1..n} gcd(k,n)^(k-1).

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A342389 a(n) = Sum_{k=1..n} k^gcd(k,n).

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Mathematica

PARI

Formula

A342396 a(n) = Sum_{k=1..n} k^(n/gcd(k,n) - 1).

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Author

Keywords

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Mathematica

PARI

Formula