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 1-6 of 6 results.

A342471 a(n) = Sum_{d|n} phi(d)^n.

Original entry on oeis.org

1, 2, 9, 18, 1025, 130, 279937, 65794, 10078209, 2097154, 100000000001, 16789506, 106993205379073, 156728328194, 35185445863425, 281479271743490, 295147905179352825857, 203119913861122, 708235345355337676357633, 1152923703631151106
Offset: 1

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^n &]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^n);
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (eulerphi(k)*x)^k/(1-(eulerphi(k)*x)^k)))

Formula

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

A164941 a(n) = Sum_{d|n} phi(n/d)^(d-1).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 6, 7, 2, 17, 2, 9, 34, 15, 2, 45, 2, 87, 102, 13, 2, 191, 258, 15, 294, 289, 2, 1579, 2, 203, 1126, 19, 5394, 2577, 2, 21, 4242, 17227, 2, 16083, 2, 2037, 83282, 25, 2, 36107, 46658, 262423, 65794, 5839, 2, 139161, 1058578, 292455
Offset: 1

Views

Author

Franklin T. Adams-Watters, Sep 01 2009

Keywords

Links

Crossrefs

Cf. A000010, A087909, A309369.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^(n/d-1)); \\ Seiichi Manyama, Mar 13 2021
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(gcd(k, n)-2)); \\ Seiichi Manyama, Mar 13 2021
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-eulerphi(k)*x^k))) \\ Seiichi Manyama, Mar 13 2021

Formula

G.f.: Sum_{k>=1} x^k/(1-phi(k)*x^k).
From Seiichi Manyama, Mar 13 2021: (Start)
a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(gcd(k, n) - 2).
If p is prime, a(p) = 2. (End)

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

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

A342544 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(gcd(k, n) - 1).

Original entry on oeis.org

1, 2, 6, 11, 260, 40, 46662, 16398, 1679630, 262408, 10000000010, 4194366, 8916100448268, 13060740684, 4398046511640, 35184372105244, 18446744073709551632, 16926661124436, 39346408075296537575442, 144115188076118572, 3833759992447475215524, 1000000000010000000020
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A309369, A342436, A342540, A342542, A342543.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^(# - 1) &]; Array[a, 20] (* Amiram Eldar, Mar 15 2021 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(gcd(k, n))^(gcd(k, n)-1));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^(d-1));

Formula

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

A344061 a(n) = Sum_{d|n} sigma(d)^(n/d).

Original entry on oeis.org

1, 4, 5, 17, 7, 56, 9, 146, 78, 298, 13, 1501, 15, 2276, 1265, 9219, 19, 25716, 21, 77519, 16929, 177328, 25, 739582, 7808, 1594562, 264382, 5611241, 31, 15699452, 33, 48863172, 4196081, 129140542, 312753, 447589422, 39, 1162261928, 67111665, 3771805472, 43, 10764897556, 45
Offset: 1

Views

Author

Seiichi Manyama, May 08 2021

Keywords

Crossrefs

Cf. A007429, A055225, A279789, A309369, A342471, A344060.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[1 , #]^(n/#) &]; Array[a, 43] (* Amiram Eldar, May 08 2021 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d)^(n/d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)*x^k/(1-sigma(k)*x^k)))

Formula

G.f.: Sum_{k >= 1} sigma(k) * x^k/(1 - sigma(k) * x^k).
If p is prime, a(p) = 2 + p.
Showing 1-6 of 6 results.

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