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.

A347084 Dirichlet inverse of A129283, n + A003415(n).

Original entry on oeis.org

1, -3, -4, 1, -6, 13, -8, 1, 1, 19, -12, -6, -14, 25, 25, 1, -18, -5, -20, -8, 33, 37, -24, -5, 1, 43, 2, -10, -30, -87, -32, 1, 49, 55, 49, 6, -38, 61, 57, -7, -42, -113, -44, -14, -8, 73, -48, -4, 1, -5, 73, -16, -54, -9, 73, -9, 81, 91, -60, 51, -62, 97, -10, 1, 85, -165, -68, -20, 97, -163, -72, 2, -74, 115
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2021

Keywords

Links

Crossrefs

Cf. A003415, A129283, A347082, A347085, A347086, A348995 (positions of 1's).
Cf. also A346241, A348976.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
A347084(n) = v347084[n];

Formula

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA129283(n/d).
a(n) = A347085(n) - A129283(n).
a(n) = A347082(n) - A347086(n).

A347082 Dirichlet inverse of -A168036, n - A003415(n).

Original entry on oeis.org

1, -1, -2, 1, -4, 3, -6, 3, 1, 5, -10, 0, -12, 7, 9, 9, -16, 1, -18, -2, 13, 11, -22, 1, 1, 13, 4, -4, -28, -13, -30, 27, 21, 17, 25, 6, -36, 19, 25, -5, -40, -19, -42, -8, -2, 23, -46, 6, 1, 1, 33, -10, -52, 5, 41, -11, 37, 29, -58, 3, -60, 31, -4, 81, 49, -31, -66, -14, 45, -33, -70, 22, -72, 37, 0, -16, 61, -37, -78
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2021

Keywords

Links

Crossrefs

Cf. A003415, A168036, A347083, A347084, A347086, A347087.
Cf. also A346241.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347082 = DirInverseCorrect(vector(up_to,n,n-A003415(n)));
A347082(n) = v347082[n];

Formula

a(1) = 1; and for n > 2, a(n) = Sum_{d|n, dA168036(n/d).
a(n) = A168036(n) + A347083(n).
a(n) = A347084(n) + A347086(n).
a(n) = A347087(n) - n.

A347085 Sum of A129283 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 21, 16, 36, 0, 22, 0, 48, 48, 49, 0, 34, 0, 36, 64, 72, 0, 63, 36, 84, 56, 50, 0, -26, 0, 113, 96, 108, 96, 102, 0, 120, 112, 101, 0, -30, 0, 78, 76, 144, 0, 156, 64, 90, 144, 92, 0, 126, 144, 139, 160, 180, 0, 203, 0, 192, 104, 257, 168, -38, 0, 120, 192, -34, 0, 230, 0, 228, 124, 134, 192
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2021

Keywords

Links

Crossrefs

Cf. A003415, A129283, A347083, A347084, A347086.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n + A003415(n));
v347084 = DirInverseCorrect(vector(up_to,n,A129283(n)));
A347084(n) = v347084[n];
A347085(n) = (A129283(n)+A347084(n));

Formula

a(n) = A129283(n) + A347084(n).
For n > 1, a(n) = -Sum_{d|n, 1A129283(d) * A347084(n/d).

A347087 Sum of n and the Dirichlet inverse of n - A003415(n).

Original entry on oeis.org

2, 1, 1, 5, 1, 9, 1, 11, 10, 15, 1, 12, 1, 21, 24, 25, 1, 19, 1, 18, 34, 33, 1, 25, 26, 39, 31, 24, 1, 17, 1, 59, 54, 51, 60, 42, 1, 57, 64, 35, 1, 23, 1, 36, 43, 69, 1, 54, 50, 51, 84, 42, 1, 59, 96, 45, 94, 87, 1, 63, 1, 93, 59, 145, 114, 35, 1, 54, 114, 37, 1, 94, 1, 111, 75, 60, 138, 41, 1, 68, 97, 123, 1, 87
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2021

Keywords

Comments

The first negative term is a(1408) = -131.

Links

Crossrefs

Cf. A003415, A168036, A347082, A347083, A347086.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347082 = DirInverseCorrect(vector(up_to,n,n-A003415(n)));
A347082(n) = v347082[n];
A347087(n) = (n+A347082(n));

Formula

a(n) = n + A347082(n).
Showing 1-4 of 4 results.

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