A342541 -id:A342541

Search: a342541 -id:a342541

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A342539

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

+10
4

1, 2, 10, 19, 1028, 132, 279942, 65798, 10078726, 2097160, 100000000010, 16797702, 106993205379084, 156728328204, 35186519703560, 281479271809036, 295147905179352825872, 203119914385420, 708235345355337676357650, 1152924803145924620, 46005163783270994804748, 20000000000000000000020

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

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

a(n) = Sum_{k=1..n} phi(n/gcd(n,k))^(n-1)*phi(gcd(n,k)). - Richard L. Ollerton, May 09 2021

MATHEMATICA

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^n &]; Array[a, 20] (* Amiram Eldar, Mar 15 2021 *)

PROG

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

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

CROSSREFS

Cf. A000010, A029935, A332517, A342487, A342534, A342535, A342540, A342541, A342543.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 15 2021

STATUS

approved

A342542

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

+10
4

1, 2, 3, 4, 5, 7, 7, 9, 15, 13, 11, 33, 13, 19, 105, 33, 17, 91, 19, 209, 469, 31, 23, 641, 1045, 37, 1627, 841, 29, 4217, 31, 673, 10461, 49, 29785, 10281, 37, 55, 49465, 68769, 41, 65197, 43, 12281, 529625, 67, 47, 273185, 279979, 1049661, 1049121, 52657, 53, 803647

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

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

MATHEMATICA

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^(n/#-1) &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)

PROG

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

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

CROSSREFS

Cf. A000010, A029935, A342473, A342540, A342541, A342544.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 15 2021

STATUS

approved

A342543

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

+10
4

1, 2, 10, 19, 1028, 76, 279942, 65558, 10077718, 1049608, 100000000010, 16777334, 106993205379084, 78364444044, 35184372090920, 281474976776236, 295147905179352825872, 101559966746268, 708235345355337676357650, 1152921504607897676, 46005119909369702026044, 10000000000100000000020

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

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

MATHEMATICA

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^# &]; Array[a, 20] (* Amiram Eldar, Mar 15 2021 *)

PROG

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

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

CROSSREFS

Cf. A000010, A029935, A342423, A342488, A342539, A342541, A342544.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 15 2021

STATUS

approved

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