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.

A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

Original entry on oeis.org

1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 1024, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1
Offset: 1

Views

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

Comments

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Crossrefs

Cf. A001227.
Cf. A318453 (numerators), A318455.

Programs

  • Mathematica
    f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
    Table[f[n] // Denominator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)
  • PARI
    up_to = 16384;
    A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
    DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
    v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
    A318454(n) = denominator(v318453_54[n]);

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318455(n).
Sum_{k=1..n} A318453(k) / a(k) ~ n/sqrt(2). - Vaclav Kotesovec, May 09 2025

A318455 The 2-adic valuation of A318454(n).

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 8, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 10, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 8, 0, 1, 0, 3, 0, 1, 0, 4, 0
Offset: 1

Views

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

Crossrefs

Programs

  • Mathematica
    f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
    a[n_] := IntegerExponent[Denominator[f[n]], 2];
    Array[a, 105] (* Jean-François Alcover, Sep 13 2018 *)
  • PARI
    A318455(n) = valuation(A318454(n),2); \\ Needs also program from A318454.

Formula

a(n) = A007814(A318454(n)).

A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 231, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 3, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1
Offset: 1

Views

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

Crossrefs

Cf. A068068, A318314 (denominators).
Differs from A318453 for the first time at n=81, where a(81) = 3, while A318453(81) = 1.

Programs

  • PARI
    up_to = 16384;
    A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
    DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
    v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
    A318313(n) = numerator(v318313_15[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d 1.
Sum_{k=1..n} A318313(k) / A318314(k) ~ 2*n/Pi. - Vaclav Kotesovec, May 10 2025
Showing 1-3 of 3 results.