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.

A304575 a(n) = Sum_{d|n} #{k < d, k squarefree and relatively prime to d}.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 7, 8, 8, 12, 9, 12, 12, 15, 12, 16, 13, 17, 16, 18, 16, 24, 17, 20, 20, 23, 18, 26, 20, 28, 23, 26, 24, 33, 24, 29, 28, 35, 27, 35, 29, 37, 34, 35, 31, 46, 32, 38, 35, 41, 33, 45, 36, 47, 38, 42, 37, 54, 38, 46, 46, 54, 42, 53, 42, 54
Offset: 1

Views

Author

Gus Wiseman, May 14 2018

Keywords

Comments

Note that a(n) <= n.

Links

Crossrefs

Cf. A000010, A001221, A005117, A008683, A073311, A111972, A112399, A139555, A265675, A304573, A304575, A304576.

Programs

  • Mathematica
    s[n_]:=Length[Select[Range[n],And[SquareFreeQ[#],GCD[n,#]===1]&]];
Table[DivisorSum[n,s],{n,100}]
  • PARI
    a(n) = sumdiv(n, d, #select(k->(issquarefree(k) && (gcd(k, d)==1)), [1..d])); \\ Michel Marcus, May 15 2018

Formula

a(n) = Sum_{d|n} A073311(d) (inverse Moebius transform of A073311). - Amiram Eldar, Nov 21 2024

A304576 a(n) = Sum_{k < n, k squarefree and relatively prime to n} (-1)^(k-1).

Original entry on oeis.org

1, 1, 0, 2, 1, 2, 1, 4, 2, 3, 1, 4, 2, 5, 2, 7, 3, 6, 4, 7, 4, 9, 5, 8, 5, 10, 3, 9, 5, 8, 5, 13, 5, 13, 5, 11, 7, 15, 5, 14, 8, 11, 8, 17, 6, 18, 8, 15, 8, 17, 7, 19, 10, 16, 9, 20, 9, 23, 12, 15, 13, 25, 8, 26, 10, 18, 13, 26, 11, 22, 14, 22, 15, 30, 9, 29
Offset: 1

Views

Author

Gus Wiseman, May 14 2018

Keywords

Crossrefs

Cf. A000010, A001221, A005117, A008683, A073311, A111972, A112399, A139555, A265675, A304573, A304574, A304575.

Programs

  • Mathematica
    l[n_]:=Sum[(-1)^(k-1),{k,Select[Range[n],And[SquareFreeQ[#],GCD[n,#]==1]&]}];
Table[l[n],{n,100}]
  • PARI
    a(n) = sum(k=1, n, if (issquarefree(k) && (gcd(n,k)==1), (-1)^(k-1))); \\ Michel Marcus, May 15 2018

A332685 a(n) = Sum_{k=1..n} mu(k/gcd(n, k)).

Original entry on oeis.org

1, 2, 1, 2, 0, 2, 0, 0, -1, 0, 0, 0, -1, -2, -2, -3, 0, -4, -1, -5, -4, -2, 0, -8, -3, -4, -4, -7, 0, -8, -2, -10, -5, -4, -4, -13, 0, -5, -4, -13, 1, -15, -1, -9, -10, -5, -1, -22, -4, -12, -5, -11, -1, -19, -6, -17, -6, -4, 1, -28, 0, -8, -12, -18, -6, -19, 0, -12, -5, -17
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2020

Keywords

Comments

Inverse Moebius transform of A112399.

Links

Crossrefs

Cf. A002321, A007431, A007947, A008683, A063659, A112399, A276833, A304575.

Programs

  • Mathematica
    Table[Sum[MoebiusMu[k/GCD[n, k]], {k, 1, n}], {n, 1, 70}]
  • PARI
    a(n) = sum(k=1, n, moebius(k/gcd(n, k))); \\ Michel Marcus, Feb 21 2020

Formula

a(n) = Sum_{k=1..n} mu(lcm(n, k)/n).
a(n) = Sum_{d|n} A112399(d).

A128431 Triangle read by rows: A054521 * A128407.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, 0, -1, 0, 1, -1, -1, 0, 0, 1, 0, 0, 0, -1, 0, 1, -1, -1, 0, -1, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, -1, 0, 0, -1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0
Offset: 1

Views

Author

Gary W. Adamson, Mar 03 2007

Keywords

Comments

Row sums = A112399: (1, 1, 0, 0, -1, 0, -1, -2, -2, -1, ...).

Examples

			First few rows of the triangle:
  1;
  1,  0;
  1, -1,  0;
  1,  0, -1,  0;
  1, -1, -1,  0,  0;
  1,  0,  0,  0, -1,  0;
  1, -1, -1,  0, -1,  1,  0;

Crossrefs

Cf. A054521, A128407, A112399, A008683.

Formula

A054521 * A128407 as infinite lower triangular matrices.
Showing 1-4 of 4 results.

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

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