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 1-3 of 3 results.

A343465 a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-3)^d.

Original entry on oeis.org

3, -3, 11, -21, 51, -119, 315, -831, 2195, -5883, 16107, -44357, 122643, -341487, 956635, -2690841, 7596483, -21522347, 61171659, -174342165, 498112275, -1426403751, 4093181691, -11767920107, 33891544419, -97764009003, 282429537947, -817028472645, 2366564736723, -6863037262207
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 16 2021

Keywords

Crossrefs

Cf. A000010, A001867, A032308, A038064, A074763, A261582, A343466, A343467.

Programs

  • Mathematica
    Table[-(1/n) Sum[EulerPhi[n/d] (-3)^d, {d, Divisors[n]}], {n, 1, 30}]
nmax = 30; CoefficientList[Series[Sum[EulerPhi[k] Log[1 + 3 x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} phi(k) * log(1 + 3*x^k) / k.
a(n) = -(1/n) * Sum_{k=1..n} (-3)^gcd(n,k).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A032308.
Product_{n>=1} (1 - x^n)^a(n) = g.f. for A261582.

A343466 a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-4)^d.

Original entry on oeis.org

4, -6, 24, -66, 208, -676, 2344, -8226, 29144, -104760, 381304, -1398476, 5162224, -19172796, 71582944, -268439586, 1010580544, -3817734596, 14467258264, -54975633768, 209430787824, -799644629556, 3059510616424, -11728124734476, 45035996273872, -173215367702376, 667199944815064
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 16 2021

Keywords

Crossrefs

Cf. A000010, A001868, A038065, A074763, A261568, A343465, A343467.

Programs

  • Mathematica
    Table[-(1/n) Sum[EulerPhi[n/d] (-4)^d, {d, Divisors[n]}], {n, 1, 27}]
nmax = 27; CoefficientList[Series[Sum[EulerPhi[k] Log[1 + 4 x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} phi(k) * log(1 + 4*x^k) / k.
a(n) = -(1/n) * Sum_{k=1..n} (-4)^gcd(n,k).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A261568.

A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.

Original entry on oeis.org

1, 2, 0, 3, -1, 1, 4, -3, 4, 0, 5, -6, 11, -4, 1, 6, -10, 24, -21, 8, 0, 7, -15, 45, -66, 51, -10, 1, 8, -21, 76, -160, 208, -119, 20, 0, 9, -28, 119, -330, 629, -676, 315, -34, 1, 10, -36, 176, -609, 1560, -2590, 2344, -831, 60, 0, 11, -45, 249, -1036, 3367, -7750, 11165, -8226, 2195, -100, 1
Offset: 1

Views

Author

Seiichi Manyama, Apr 11 2025

Keywords

Examples

			Square array begins:
  1,   2,    3,    4,     5,     6,      7, ...
  0,  -1,   -3,   -6,   -10,   -15,    -21, ...
  1,   4,   11,   24,    45,    76,    119, ...
  0,  -4,  -21,  -66,  -160,  -330,   -609, ...
  1,   8,   51,  208,   629,  1560,   3367, ...
  0, -10, -119, -676, -2590, -7750, -19565, ...
  1,  20,  315, 2344, 11165, 39996, 117655, ...

Crossrefs

Columns k=1..5 give A000035, (-1)^(n+1) * A074763(n), A343465, A343466, A343467.
Main diagonal gives A382998.
Cf. A000010, A075195, A286957, A382994, A383011.

Programs

  • PARI
    a(n, k) = -sumdiv(n, d, eulerphi(n/d)*(-k)^d)/n;

Formula

A(n,k) = (1/n) * A382994(n,k).
A(n,k) = -(1/n) * Sum_{j=1..n} (-k)^gcd(n,j).
G.f. of column k: Sum_{j>=1} phi(j) * log(1 + k*x^j) / j.
Product_{n>=1} 1/(1 - x^n)^A(n,k) = Product_{n>=1} (1 + k*x^n).
Showing 1-3 of 3 results.

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