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 1-6 of 6 results.

A308116 Sum of the perimeters of the integer-sided triangles with perimeter n whose side lengths are squarefree.

Original entry on oeis.org

0, 0, 3, 0, 5, 6, 14, 8, 9, 0, 22, 12, 39, 28, 75, 48, 51, 36, 38, 20, 63, 66, 138, 96, 75, 52, 162, 112, 203, 210, 341, 256, 264, 204, 385, 288, 333, 228, 507, 360, 410, 252, 559, 396, 720, 552, 940, 720, 833, 600, 765, 520, 848, 756, 825, 616, 684, 580
Offset: 1

Views

Author

Wesley Ivan Hurt, May 13 2019

Keywords

Links

Crossrefs

Cf. A008683, A308061.

Programs

  • Mathematica
    Table[n*Sum[Sum[MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - i - k]^2*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = n * A308061(n).
a(n) = n * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Möbius function (A008683).

A308424 Number of integer-sided triangles with perimeter n whose side lengths are nonsquarefree.

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, May 26 2019

Keywords

Links

Crossrefs

Cf. A008683, A308061.

Programs

  • Mathematica
    Table[Sum[Sum[(1 - MoebiusMu[i]^2)*(1 - MoebiusMu[k]^2)*(1 - MoebiusMu[n - i - k]^2)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (1 - mu(i)^2) * (1 - mu(k)^2) * (1 - mu(n-i-k)^2), where mu is the Möbius function (A008683).

A308103 Number of scalene triangles with perimeter n whose side lengths are squarefree.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 3, 2, 0, 1, 2, 2, 4, 5, 7, 5, 3, 4, 6, 4, 4, 5, 7, 5, 4, 5, 7, 5, 9, 11, 13, 10, 9, 10, 9, 6, 10, 11, 10, 7, 6, 8, 9, 8, 12, 16, 17, 14, 13, 16, 18, 13, 22, 24, 27, 20, 19, 23, 24, 22, 29, 34
Offset: 1

Views

Author

Wesley Ivan Hurt, May 12 2019

Keywords

Links

Crossrefs

Cf. A008683, A308061.

Programs

  • Mathematica
    Table[Sum[Sum[MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - i - k]^2*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Möbius function (A008683).

A308140 Sum of the largest side lengths of all integer-sided triangles with squarefree side lengths and perimeter n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 6, 3, 3, 0, 10, 5, 17, 12, 32, 20, 20, 13, 14, 7, 27, 30, 64, 43, 32, 21, 71, 48, 92, 92, 154, 112, 110, 85, 169, 123, 142, 94, 222, 154, 171, 101, 245, 169, 316, 250, 424, 321, 361, 263, 322, 219, 367, 337, 348, 260, 275, 242, 405, 310
Offset: 1

Views

Author

Wesley Ivan Hurt, May 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A308061, A308116.

Programs

  • Maple
    N:= 100: # for a(1)..a(N)
SF:= select(numtheory:-issqrfree, [$1..N/2]):
V:= Vector(N):
for ia from 1 to nops(SF) do
    a:= SF[ia];
    if 2*a >= N then break fi;
    for ib from ia by -1 to 1 do
      b:= SF[ib];
      if 2*b <= a then break fi;
      cs:= select(c -> b+c > a, SF[1...ib]);
      P:= select(`<=`,map(c -> a+b+c, cs),N);
      V[P]:= V[P] +~ a;
od od:
convert(V,list); # Robert Israel, May 14 2019
  • Mathematica
    Table[Sum[Sum[(n - i - k)* MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - k - i]^2 *Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
  • PARI
    a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, sign((i+k)\(n-i-k+1))* issquarefree(i)*issquarefree(k)*issquarefree(n-i-k)*(n-i-k))); \\ Michel Marcus, May 14 2019

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(i)^2 * mu(k)^2 * mu(n-i-k)^2 * (n-i-k), where mu is the Möbius function (A008683).

A335621 Number of integer-sided triangles with perimeter n such that the sum of each pair of side lengths is squarefree.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 1, 2, 3, 4, 2, 3, 2, 2, 4, 5, 4, 4, 3, 3, 3, 3, 5, 5, 4, 4, 6, 8, 5, 8, 8, 12, 7, 12, 7, 10, 6, 10, 10, 15, 14, 20, 14, 18, 17, 21, 20, 25, 20, 23, 18, 19, 16, 20, 22, 24, 21, 25, 21, 22, 20, 22, 23, 28, 22, 28, 22, 24, 20, 23, 25
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 02 2020

Keywords

Links

Crossrefs

Cf. A008683, A308061 (each side length is squarefree).
Cf. A335621 (averages of each pair of side lengths is prime).

Programs

  • Mathematica
    Table[Sum[Sum[MoebiusMu[i + k]^2*MoebiusMu[n - i]^2*MoebiusMu[n - k]^2 * Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(i+k)^2 * mu(n-i)^2 * mu(n-k)^2, where mu is the Möbius function (A008683).

A307987 Number of integer-sided triangles with perimeter n and at least one nonsquarefree side.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 8, 7, 9, 7, 8, 8, 13, 12, 13, 12, 14, 12, 13, 13, 19, 18, 19, 19, 24, 24, 24, 24, 30, 31, 31, 31, 32, 32, 32, 33, 39, 40, 46, 46, 49, 47, 55, 54, 63, 60, 64, 63, 67, 62, 66, 66, 75, 72, 76, 77, 79, 75, 79
Offset: 1

Views

Author

Wesley Ivan Hurt, May 18 2019

Keywords

Links

Crossrefs

Cf. A005044, A008683, A308061.

Programs

  • Mathematica
    Table[Sum[Sum[(1 - MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - i - k]^2)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (1 - mu(i)^2 * mu(k)^2 * mu(n-i-k)^2), where mu is the Möbius function (A008683).
a(n) = A005044(n) - A308061(n).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.