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.

A307776 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+1)*a(d).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 0, 1, -2, -1, 0, 1, 0, 1, 1, 1, -3, -2, 0, 1, 2, 3, 3, 4, 1, 1, 1, 0, -2, -1, -2, -1, -6, -5, -2, -2, 2, 3, 3, 4, 3, 4, 3, 4, 0, -1, -4, -3, -11, -10, -11, -13, -15, -14, -15, -15, -18, -17, -15, -14, -11, -10, -8, -7, -11, -11, -2, -1, 6, 10, 13, 14, 21, 22, 20
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A003238, A281487, A307777, A307778, A307779.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,d,n; a:=[1]:
for n from 1 to q do a:=[op(a),add((-1)^(n/d+1)*a[d],d=divisors(n))]:
od; op(a); end: P(74); # Paolo P. Lava, Apr 30 2019
  • Mathematica
    a[n_] := a[n] = Sum[(-1)^((n - 1)/d + 1) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 75}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[a[k] x^k/(1 + x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 75}]

Formula

G.f.: x * (1 + Sum_{n>=1} a(n)*x^n/(1 + x^n)).
L.g.f.: log(Product_{n>=1} (1 + x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A307778 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(d+1)*a(d).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 0, 1, -2, -1, 0, 1, -2, -1, 1, 1, -3, -2, 0, 1, -2, -1, 1, 2, -5, -5, 3, 2, -2, -1, 2, 3, -6, -5, 2, 2, -4, -3, 3, 2, -5, -4, 3, 4, -4, -5, 6, 7, -13, -12, 7, 5, -3, -2, 5, 5, -8, -7, 5, 6, -7, -6, 8, 5, -11, -13, 8, 9, -8, -6, 9, 10, -17, -16, 12, 8, -6
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A003238, A281487, A307776, A307777, A307779.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[(-1)^(d + 1) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 77}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[(-1)^(k + 1) a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 77}]

Formula

G.f.: x * (1 + Sum_{n>=1} (-1)^(n+1)*a(n)*x^n/(1 - x^n)).
L.g.f.: log(Product_{n>=1} (1 - x^n)^((-1)^n*a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A307777 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+d)*a(d).

Original entry on oeis.org

1, 1, -2, -1, 1, 2, -2, -1, 0, -1, -2, -1, 6, 7, -7, -7, 5, 6, -8, -7, 6, 3, -3, -2, 5, 7, -15, -16, 26, 27, -22, -21, 12, 9, -16, -16, 28, 29, -23, -18, 9, 10, -23, -22, 28, 21, -20, -19, 25, 24, -31, -27, 29, 30, -23, -23, 16, 7, -35, -34, 79, 80, -60, -57, 27, 35, -50, -49, 54, 50, -41
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A003238, A281487, A307776, A307778, A307779.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[(-1)^((n - 1)/d + d) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 71}]
a[n_] := a[n] = SeriesCoefficient[x (1 - Sum[a[k] (-x)^k/(1 + x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 71}]

Formula

G.f.: x * (1 - Sum_{n>=1} a(n)*(-x)^n/(1 + x^n)).
L.g.f.: log(Product_{n>=1} (1 + x^n)^((-1)^(n+1)*a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A335283 a(n) = 1 + Sum_{d|n, n/d odd, d < n} a(d).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 6, 1, 2, 4, 2, 2, 6, 2, 2, 2, 4, 2, 8, 2, 2, 6, 2, 1, 6, 2, 6, 4, 2, 2, 6, 2, 2, 6, 2, 2, 16, 2, 2, 2, 4, 4, 6, 2, 2, 8, 6, 2, 6, 2, 2, 6, 2, 2, 16, 1, 6, 6, 2, 2, 6, 6, 2, 4, 2, 2, 16, 2, 6, 6, 2, 2, 16, 2, 2, 6, 6, 2, 6, 2, 2, 16, 6, 2, 6, 2, 6, 2, 2, 4, 16, 4, 2, 6, 2, 2, 26
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2020

Keywords

Links

Crossrefs

Cf. A000079 (positions of 1's), A038550 (positions of 2's), A067824, A074206, A209229, A307779, A335062.

Programs

  • Mathematica
    a[n_] := 1 + Sum[If[d < n && OddQ[n/d], a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 85}]
terms = 85; A[] = 0; Do[A[x] = x/(1 - x) + Sum[A[x^(2 k - 1)], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest
  • PARI
    up_to = 20000;
A335283lista(up_to) = {my(v = vector(up_to)); for(n=1, up_to, v[n] = 1 + sumdiv(n, d, if(dA335283lista(up_to);
A335283(n) = v335283[n]; \\ Antti Karttunen, Dec 09 2021

Formula

G.f. A(x) satisfies: A(x) = x / (1 - x) + Sum_{k>=2} A(x^(2*k-1)).
G.f.: x / (1 - x) + Sum_{n>=1} a(n) * x^(3*n) / (1 - x^(2*n)).
a(1) = 1; a(2*n) = a(n), a(2*n+1) = 2 * A074206(2*n+1).

Extensions

More terms from Antti Karttunen, Dec 09 2021

A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 7, 7, 8, 8, 10, 10, 10, 10, 12, 13, 13, 14, 16, 16, 16, 16, 19, 19, 19, 19, 24, 24, 24, 24, 28, 28, 28, 28, 32, 34, 34, 34, 39, 40, 41, 41, 46, 46, 48, 48, 53, 53, 53, 53, 58, 58, 58, 60, 67, 67, 67, 67, 74, 74, 74, 74, 84, 84, 84, 85, 93, 93, 93, 93
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 28 2025

Keywords

Crossrefs

Cf. A003238, A076752, A167865, A307779, A317240.

Programs

  • Maple
    a:= proc(n) option remember; uses numtheory; `if`(n=1, 1,
      add(`if`(issqr(d), a((n-1)/d), 0), d=divisors(n-1)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 28 2025
  • Mathematica
    a[1] = 1; a[n_] := a[n] = DivisorSum[n - 1, a[(n - 1)/#] &, IntegerQ[Sqrt[#]] &]; Table[a[n], {n, 1, 80}]
nmax = 80; A[] = 0; Do[A[x] = x (1 + Sum[A[x^(k^2)], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * (1 + A(x) + A(x^4) + A(x^9) + ... + A(x^(k^2)) + ...).
Showing 1-5 of 5 results.

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