A342608 -id:A342608 - cp's OEIS frontend

A342607 a(n) = Sum_{d|n} phi(d)^(n-d).

1, 2, 2, 3, 2, 11, 2, 19, 66, 1027, 2, 835, 2, 279939, 1052674, 69635, 2, 10114563, 2, 1074855939, 78364426242, 100000000003, 2, 4315152387, 1099511627778, 106993205379075, 101559973445634, 21937029021319171, 2, 1162183941554179, 2, 562950221856771, 10000000000001073741826

Offset: 1

Seiichi Manyama, Mar 16 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..605

Cf. A000010, A164941, A342608, A342612, A342628.

a[n_] := DivisorSum[n, EulerPhi[#]^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 17 2021 *)

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

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

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

a(n) = Sum_{k=1..n} phi(n/gcd(k,n))^(n - n/gcd(k,n) - 1).
G.f.: Sum_{k>=1} x^k/(1 - (phi(k) * x)^k).
If p is prime, a(p) = 2.

A342613 a(n) = Sum_{d|n} phi(n/d)^(n+d).

Original entry on oeis.org

1, 2, 17, 34, 4097, 386, 1679617, 263170, 60470273, 20971522, 1000000000001, 67223554, 1283918464548865, 3291294892034, 281543697235969, 2251868534210562, 4722366482869645213697, 4265518198947842, 12748236216396078174437377, 9223671104051085314, 552066177293775007645697, 1100000000000000000000002

Offset: 1

Seiichi Manyama, Mar 16 2021

nonn

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

Cf. A342608, A342612, A342620.

a[n_] := DivisorSum[n, EulerPhi[n/#]^(n + #) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)

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

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

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

a(n) = Sum_{k=1..n} phi(n/gcd(k,n))^(n + gcd(k,n) - 1).
G.f.: Sum_{k>=1} phi(k)^(k+1) * x^k/(1 - phi(k)^(k+1) * x^k).
If p is prime, a(p) = 1 + (p-1)^(p+1).

A342618 a(n) = Sum_{d|n} phi(d)^(n+d+1).

Original entry on oeis.org

1, 2, 129, 514, 4194305, 9218, 470184984577, 17179877378, 609359740018689, 4402341478402, 100000000000000000000001, 1125899907563522, 137370551967459378662586974209, 36845784908492735250434, 9903520314283046597240029185

Offset: 1

Seiichi Manyama, Mar 16 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..220

Cf. A000010, A342608.

a[n_] := DivisorSum[n, EulerPhi[#]^(n + # + 1) &]; Array[a, 15] (* Amiram Eldar, Mar 17 2021 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^(2*k+1)*x^k/(1-(eulerphi(k)*x)^k)))

a(n) = Sum_{k=1..n} phi(n/gcd(k,n))^(n + n/gcd(k,n)).
G.f.: Sum_{k>=1} phi(k)^(2*k+1) * x^k/(1 - (phi(k) * x)^k).
If p is prime, a(p) = 1 + (p-1)^(2*p+1).

cp's OEIS Frontend

A342607 a(n) = Sum_{d|n} phi(d)^(n-d).

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A342613 a(n) = Sum_{d|n} phi(n/d)^(n+d).

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A342618 a(n) = Sum_{d|n} phi(d)^(n+d+1).

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