A342473 -id:A342473 - cp's OEIS frontend

A342485 a(n) = Sum_{d|n} phi(d)^(d+1).

1, 2, 17, 34, 4097, 146, 1679617, 262178, 60466193, 4198402, 1000000000001, 67109042, 1283918464548865, 470186664194, 281474976714769, 2251799813947426, 4722366482869645213697, 609359800476818, 12748236216396078174437377, 9223372036858974242

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

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

Cf. A000010, A342473, A342487, A342488, A342489.

a[n_] := DivisorSum[n, EulerPhi[#]^(#+1) &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)

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

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

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

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n/gcd(k, n)).
G.f.: Sum_{k>=1} phi(k)^(k+1) * x^k/(1 - x^k).
If p is prime, a(p) = 1 + (p-1)^(p+1).
a(n) = Sum_{k=1..n} phi(gcd(n,k))^(gcd(n,k) + 1)/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A342542 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n) - 1).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

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

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

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

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

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

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

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

A344047 a(n) = Sum_{d|n} sigma(d)^d.

Original entry on oeis.org

1, 10, 65, 2411, 7777, 2986058, 2097153, 2562893036, 10604499438, 3570467234410, 743008370689, 232218265092200875, 793714773254145, 21035720123170684938, 504857282956046114465, 727423121747187826721517, 2185911559738696531969, 43567528752021332763809905512, 5242880000000000000000001

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A065018, A342473, A344044.

a[n_] := DivisorSum[n, DivisorSigma[1, #]^# &]; Array[a, 19] (* Amiram Eldar, May 08 2021 *)

PARI
```
a(n) = sumdiv(n, d, sigma(d)^d);
```

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

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

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

A342489 a(n) = Sum_{d|n} phi(d)^(d-1).

Original entry on oeis.org

1, 2, 5, 10, 257, 38, 46657, 16394, 1679621, 262402, 10000000001, 4194350, 8916100448257, 13060740674, 4398046511365, 35184372105226, 18446744073709551617, 16926661124390, 39346408075296537575425, 144115188076118282, 3833759992447475168837

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

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

Cf. A000010, A014566, A164941, A342473, A342485.

a[n_] := DivisorSum[n, EulerPhi[#]^(#-1) &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)

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

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

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

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n/gcd(k, n) - 2).
G.f.: Sum_{k>=1} phi(k)^(k-1) * x^k/(1 - x^k).
If p is prime, a(p) = 1 + (p-1)^(p-1) = A014566(p-1).

cp's OEIS Frontend

A342485 a(n) = Sum_{d|n} phi(d)^(d+1).

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A342542 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n) - 1).

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A344047 a(n) = Sum_{d|n} sigma(d)^d.

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A342489 a(n) = Sum_{d|n} phi(d)^(d-1).

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