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 1-4 of 4 results.

A309369 a(n) = Sum_{d|n} phi(n/d)^d, where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 10, 15, 22, 11, 34, 13, 44, 105, 42, 17, 116, 19, 314, 357, 112, 23, 426, 1045, 158, 747, 1474, 29, 5290, 31, 594, 3069, 274, 24185, 6082, 37, 344, 9945, 67922, 41, 63542, 43, 12170, 303225, 508, 47, 74834, 279979, 1050022, 135201, 29098, 53, 309872, 4294345
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 25 2019

Keywords

Links

Crossrefs

Cf. A000010, A055225, A164941, A264782, A279789.

Programs

  • Mathematica
    Table[Sum[EulerPhi[n/d]^d, {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 55; CoefficientList[Series[-Log[Product[(1 - EulerPhi[k] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(gcd(k, n)-1)); \\ Seiichi Manyama, Mar 13 2021

Formula

G.f.: Sum_{k>=1} phi(k)*x^k/(1 - phi(k)*x^k).
L.g.f.: -log(Product_{k>=1} (1 - phi(k)*x^k)^(1/k)).
a(p) = p for p prime.
a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(gcd(k, n) - 1). - Seiichi Manyama, Mar 13 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

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).

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Mar 16 2021

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^(n-d));
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n-n/gcd(k, n)-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(eulerphi(k)*x)^k)))

Formula

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.
Showing 1-4 of 4 results.

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