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 1-7 of 7 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).

A342470 a(n) = Sum_{d|n} phi(d)^4.

Original entry on oeis.org

1, 2, 17, 18, 257, 34, 1297, 274, 1313, 514, 10001, 306, 20737, 2594, 4369, 4370, 65537, 2626, 104977, 4626, 22049, 20002, 234257, 4658, 160257, 41474, 106289, 23346, 614657, 8738, 810001, 69906, 170017, 131074, 333329, 23634, 1679617, 209954, 352529, 70418
Offset: 1

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Links

Crossrefs

Cf. A000010, A013663, A029939, A338997, A342471.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^4 &]; Array[a, 40] (* Amiram Eldar, Mar 13 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^4);
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^3);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^4*x^k/(1-x^k)))

Formula

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^3.
G.f.: Sum_{k>=1} phi(k)^4 * x^k/(1 - x^k).
From Amiram Eldar, Nov 13 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p-1)^3*(p^(4*e)-1))/(p^3 + p^2 + p + 1).
Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.05936545607... . (End)

A342540 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n-1).

Original entry on oeis.org

1, 2, 6, 11, 260, 68, 46662, 16518, 1680134, 524296, 10000000010, 4204550, 8916100448268, 26121388044, 4398583447560, 35185445896204, 18446744073709551632, 33853319413772, 39346408075296537575442, 144116012711673868, 3833767304764361539596, 2000000000000000000020
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A342471, A342539, A342542, A342544.

Programs

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

Formula

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

A344060 a(n) = Sum_{d|n} sigma(d)^n.

Original entry on oeis.org

1, 10, 65, 2483, 7777, 2990810, 2097153, 2568661988, 10604761518, 3570527751850, 743008370689, 232227195048256531, 793714773254145, 21035724521219881850, 504857283427304833025, 727429690188773950335429, 2185911559738696531969, 43567528891100073055151954340, 5242880000000000000000001
Offset: 1

Views

Author

Seiichi Manyama, May 08 2021

Keywords

Crossrefs

Cf. A007429, A023887, A065018, A342471, A344044, A344047, A344061.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[1 , #]^n &]; Array[a, 19] (* Amiram Eldar, May 08 2021 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d)^n);
  • PARI
    my(N=20, 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) = 1 + (p+1)^p.

A342490 a(n) = Sum_{d|n} phi(d)^(n-1).

Original entry on oeis.org

1, 2, 5, 10, 257, 66, 46657, 16514, 1679873, 524290, 10000000001, 4200450, 8916100448257, 26121388034, 4398314962945, 35185445863426, 18446744073709551617, 33853319151618, 39346408075296537575425, 144115737832194050, 3833763648605916233729, 2000000000000000000002
Offset: 1

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Links

Crossrefs

Cf. A000010, A014566, A342471, A342487.

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-2));
  • 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-2).
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) = A014566(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 1-7 of 7 results.

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