A342412 -id:A342412 - cp's OEIS frontend

A342432 a(n) = Sum_{k=1..n} gcd(k,n)^(n-2).

1, 2, 5, 22, 129, 1411, 16813, 266372, 4787349, 100391653, 2357947701, 61980047702, 1792160394049, 56707753687079, 1946197516142925, 72061992621375496, 2862423051509815809, 121441389759089405193, 5480386857784802185957

Offset: 1

Seiichi Manyama, Mar 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A000010, A332517, A342412, A342433, A343510.

a[n_] := Sum[GCD[k, n]^(n - 2), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^(n-2));
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n-2));

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n-3));

a(n) = Sum_{d|n} phi(n/d) * d^(n-2).
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-3)(d).
a(n) ~ n^(n-2). - Vaclav Kotesovec, May 23 2021

A342411 a(n) = Sum_{k=1..n} (n/gcd(k,n))^(n/gcd(k,n) - 2).

Original entry on oeis.org

1, 2, 7, 34, 501, 2600, 100843, 1048610, 28697821, 400000502, 23579476911, 247669459528, 21505924728445, 340163474352620, 15569560546875507, 576460752304472098, 45798768824157052689, 728637186579594211070, 98646963440126439346903

Offset: 1

Seiichi Manyama, Mar 11 2021

nonn

Cf. A000010, A226459, A342412.

a[n_] := Sum[(n/GCD[k, n])^(n/GCD[k, n] - 2), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 11 2021 *)

a(n) = sum(k=1, n, (n/gcd(k, n))^(n/gcd(k, n)-2));

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

a(n) = sumdiv(n, d, eulerphi(d)*d^(d-2));

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k^(k-1))*x^k/(1-x^k)))

a(n) = Sum_{d|n} phi(d^(d-1)) = Sum_{d|n} phi(d) * d^(d-2).

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

cp's OEIS Frontend

A342432 a(n) = Sum_{k=1..n} gcd(k,n)^(n-2).

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A342411 a(n) = Sum_{k=1..n} (n/gcd(k,n))^(n/gcd(k,n) - 2).

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Author

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Mathematica

PARI

PARI

PARI

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Formula