cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A342473 a(n) = Sum_{d|n} phi(d)^d.

Original entry on oeis.org

1, 2, 9, 18, 1025, 74, 279937, 65554, 10077705, 1049602, 100000000001, 16777306, 106993205379073, 78364444034, 35184372089865, 281474976776210, 295147905179352825857, 101559966746186, 708235345355337676357633, 1152921504607896594, 46005119909369701746057, 10000000000100000000002
Offset: 1

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Links

Crossrefs

Cf. A000010, A029939, A110567, A309369, A338997, A342470, A342471, A342485.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^# &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^d);
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n/gcd(k, n)-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (eulerphi(k)*x)^k/(1-x^k)))

Formula

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n/gcd(k, n) - 1).
G.f.: Sum_{k>=1} (phi(k) * x)^k/(1 - x^k).
If p is prime, a(p) = 1 + (p-1)^p = A110567(p-1).
a(n) = Sum_{k=1..n} phi(gcd(n,k))^gcd(n,k)/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A342487 a(n) = Sum_{d|n} phi(d)^(n+1).

Original entry on oeis.org

1, 2, 17, 34, 4097, 258, 1679617, 262658, 60467201, 8388610, 1000000000001, 67133442, 1283918464548865, 940369969154, 281479271743489, 2251816993685506, 4722366482869645213697, 1218719481069570, 12748236216396078174437377, 9223380832949895170
Offset: 1

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Links

Crossrefs

Cf. A000010, A342471, A342485, A342488, A342490.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^(n+1) &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^(n+1));
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^n);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^(k+1)*x^k/(1-(eulerphi(k)*x)^k)))

Formula

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

A342488 a(n) = Sum_{d|n} phi(d)^(n/d+1).

Original entry on oeis.org

1, 2, 5, 6, 17, 14, 37, 26, 53, 82, 101, 74, 145, 254, 385, 162, 257, 398, 325, 1218, 1697, 1102, 485, 1058, 4497, 1874, 2645, 8394, 785, 19330, 901, 2306, 14497, 4354, 112769, 17738, 1297, 6158, 37697, 270082, 1601, 316130, 1765, 105498, 1165441, 11134, 2117, 162050, 1681381
Offset: 1

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Links

Crossrefs

Cf. A000010, A002522, A164941, A309369, A342485, A342487.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^(n/#+1) &]; Array[a, 50] (* Amiram Eldar, Mar 14 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^(n/d+1));
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^gcd(k, n));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^2*x^k/(1-eulerphi(k)*x^k)))

Formula

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^gcd(k, n).
G.f.: Sum_{k>=1} phi(k)^2 * x^k/(1 - phi(k) * x^k).
If p is prime, a(p) = 1 + (p-1)^2 = A002522(p-1).
a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n) + 1)/phi(n/gcd(k, n)). - Richard L. Ollerton, May 07 2021

A342541 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n)).

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 12, 14, 28, 28, 20, 62, 24, 54, 272, 68, 32, 198, 36, 676, 1224, 130, 44, 1348, 4136, 180, 3540, 3426, 56, 12632, 60, 1640, 22520, 304, 129456, 22370, 72, 378, 101808, 270952, 80, 192996, 84, 40630, 1867184, 550, 92, 551528, 1679700, 4198860, 2105408
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A342485, A342539, A342542, A342543.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^(n/#) &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(gcd(k, n))^(n/gcd(k, n)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^(n/d));

Formula

a(n) = Sum_{d|n} phi(n/d) * phi(d)^(n/d).
If p is prime, a(p) = 2 *(p-1).

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

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^(#-1) &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^(d-1));
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n/gcd(k, n)-2));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^(k-1)*x^k/(1-x^k)))

Formula

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).
Showing 1-5 of 5 results.

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