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.

A319133 a(1) = a(2) = 1; for n > 2, a(n+2) = Sum_{d|n} tau(n/d)*a(d), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 1, 1, 3, 3, 8, 5, 16, 7, 29, 12, 41, 14, 76, 16, 92, 28, 142, 30, 185, 32, 268, 48, 298, 50, 466, 59, 500, 80, 683, 82, 817, 84, 1072, 114, 1134, 134, 1583, 136, 1649, 170, 2176, 172, 2444, 174, 3032, 239, 3134, 241, 4174, 254, 4353, 316, 5343, 318, 5815, 352, 7121, 418, 7287, 420, 9357, 422, 9527, 525
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 11 2018

Keywords

Links

Crossrefs

Cf. A000005, A003238, A007439, A007557, A318583.

Programs

  • Mathematica
    a[1] = a[2] = 1; a[n_] := a[n] = Sum[DivisorSigma[0, (n - 2)/d] a[d], {d, Divisors[n - 2]}]; Table[a[n], {n, 65}]
  • PARI
    A319133(n) = if(n<=2,1,sumdiv(n-2,d,numdiv((n-2)/d)*A319133(d))); \\ (non-memoized implementation) - Antti Karttunen, Sep 11 2018
  • PARI
    \\ Faster implementation:
up_to = 4240;
A319133list(up_to) = { my(u=vector(up_to)); u[1] = u[2] = 1; for(n=3, up_to, u[n] = sumdiv(n-2,d,numdiv((n-2)/d)*u[d])); (u); };
v319133 = A319133list(up_to);
A319133(n) = v319133[n]; \\ Antti Karttunen, Sep 11 2018

A343188 a(1) = a(2) = a(3) = 1; a(n+3) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 0, -1, -1, -1, -2, -2, -1, -3, -2, -2, -3, -2, -1, -4, 0, -2, -3, 0, 0, -4, 4, 0, -3, 5, 3, -4, 9, 2, -2, 11, 5, -1, 15, 4, 0, 16, 10, -1, 20, 9, 1, 24, 12, 0, 25, 12, 1, 28, 16, 0, 25, 19, 2, 26, 22, 1, 26, 21, -2, 28, 25, 0, 20, 24, -2, 23, 30, -3, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 07 2021

Keywords

Crossrefs

Cf. A007554, A307993, A318583, A343189, A343190.

Programs

  • Mathematica
    a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[MoebiusMu[(n - 3)/d] a[d], {d, Divisors[n - 3]}]; Table[a[n], {n, 75}]

A343189 a(1) = ... = a(4) = 1; a(n+4) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, -2, -2, -1, -2, -2, -3, -1, -3, -1, -4, 0, -3, 0, -5, 2, -4, 1, -5, 4, -5, 6, -6, 5, -4, 8, -6, 8, -5, 11, -5, 10, -6, 17, -6, 11, -3, 21, -7, 12, -3, 26, -5, 12, -4, 33, -4, 11, -1, 37, -5, 11, -2, 42, -1, 7, -1, 48, -2, 1, 3, 58, -3
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 07 2021

Keywords

Crossrefs

Cf. A007554, A307994, A318583, A343188, A343190.

Programs

  • Mathematica
    a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = Sum[MoebiusMu[(n - 4)/d] a[d], {d, Divisors[n - 4]}]; Table[a[n], {n, 75}]

A343190 a(1) = ... = a(5) = 1; a(n+5) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -2, -2, -1, -2, -1, -3, -2, -2, -2, -1, -3, -3, 0, -3, 0, -3, -2, -1, -1, -1, -2, -2, 1, -1, 3, -3, -1, 2, 2, 2, -1, -2, 5, 4, 4, -2, 1, 5, 7, 6, -1, 0, 10, 7, 10, 0, 0, 9, 14, 9, 0, 2, 12, 15, 14, -1, 3, 14, 18
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 07 2021

Keywords

Crossrefs

Cf. A007554, A307995, A318583, A343188, A343189.

Programs

  • Mathematica
    a[1] = a[2] = a[3] = a[4] = a[5] = 1; a[n_] := a[n] = Sum[MoebiusMu[(n - 5)/d] a[d], {d, Divisors[n - 5]}]; Table[a[n], {n, 75}]

A346034 a(1) = 1, a(2) = 0; a(n+2) = Sum_{d|n} mu(n/d) * a(d).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, -1, -1, -2, 0, -3, 1, -4, 3, -5, 5, -5, 6, -6, 10, -7, 11, -6, 15, -7, 14, -7, 19, -5, 17, -6, 23, -7, 18, -4, 24, -2, 16, -3, 23, 1, 13, 0, 17, -1, 7, 7, 14, 6, -7, 7, 0, 12, -13, 11, -14, 15, -33, 21, -27, 20, -57, 19, -50, 29, -73, 34, -79, 33, -96
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 01 2021

Keywords

Crossrefs

Cf. A007439, A007554, A008683, A318583, A345138, A345141.

Programs

  • Mathematica
    a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[MoebiusMu[(n - 2)/d] a[d], {d, Divisors[n - 2]}]; Table[a[n], {n, 1, 70}]
nmax = 70; A[] = 0; Do[A[x] = x + x^2 Sum[MoebiusMu[k] A[x^k], {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x + x^2 * Sum_{k>=1} mu(k) * A(x^k).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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