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).

A349338 Dirichlet convolution of A000010 (Euler totient phi) with A080339 (characteristic function of noncomposite numbers).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 7, 6, 8, 9, 11, 8, 13, 13, 14, 12, 17, 14, 19, 14, 20, 21, 23, 16, 24, 25, 24, 20, 29, 22, 31, 24, 32, 33, 34, 22, 37, 37, 38, 28, 41, 32, 43, 32, 38, 45, 47, 32, 48, 44, 50, 38, 53, 42, 54, 40, 56, 57, 59, 36, 61, 61, 54, 48, 64, 52, 67, 50, 68, 58, 71, 44, 73, 73, 68, 56, 76, 62, 79, 56, 72, 81
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Möbius transform of A230593.
The number of integers k from 1 to n such that gcd(n, k) is a noncomposite number. - Amiram Eldar, Jun 21 2025

Links

Crossrefs

Cf. A000010, A005117, A008683, A069359, A080339, A085548, A117494, A230593, A348976, A349435.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Boole[!CompositeQ[#]] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)
  • PARI
    A349338(n) = sumdiv(n, d, eulerphi(n/d)*((1==d)||isprime(d)));
  • PARI
    a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); n * vecprod(apply(x -> 1-1/x, p)) * (1 + vecsum(apply(x -> 1/x, p - vector(#e, i, e[i] == 1)~)));} \\ Amiram Eldar, Jun 21 2025

Formula

a(n) = Sum_{d|n} A000010(n/d) * A080339(d).
a(n) = Sum_{d|n} A008683(n/d) * A230593(d).
a(n) = Sum_{d|n} A349435(n/d) * A348976(d).
a(n) = A000010(n) + A117494(n). [Because A117494 is the Möbius transform of A069359]
For all n >= 1, a(A005117(n)) = A348976(A005117(n)).
Sum_{k=1..n} a(k) ~ 3 * (1 + A085548) * n^2 / Pi^2. - Vaclav Kotesovec, Nov 20 2021

A349434 Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, -2, 0, 0, 0, 6, 0, -3, 0, -2, 0, 0, 0, 0, 5, 0, 6, -2, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, -6, 7, -5, 0, -2, 0, -3, 0, 0, 0, 0, 0, 4, 0, 0, -3, 22, 0, 0, 0, -2, 0, 0, 0, -5, 0, 0, -5, -2, 0, 0, 0, -6, 21, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, -2, 0, 0, 0, -4, 0, -7, -3, 7
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Dirichlet convolution of this sequence with A349338 is A348976.

Links

Crossrefs

Cf. A003415, A129283, A230593, A349337, A349435 (Dirichlet inverse), A349436 (sum with it).
Cf. also A348976, A349338.

Programs

  • Mathematica
    s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, sinv[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    up_to = 20000;
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));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
A349337(n) = v349337[n];
A349434(n) = sumdiv(n,d,A129283(n/d)*A349337(d));

Formula

a(n) = Sum_{d|n} A129283(n/d) * A349337(d).

A349435 Dirichlet convolution of A230593 with A347084, which is Dirichlet inverse of {n + its arithmetic derivative}.

Original entry on oeis.org

1, 0, 0, -2, 0, 0, 0, -2, -3, 0, 0, 2, 0, 0, 0, -2, 0, 3, 0, 2, 0, 0, 0, 0, -5, 0, -6, 2, 0, 0, 0, -2, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, -2, -7, 5, 0, 2, 0, 3, 0, 0, 0, 0, 0, -4, 0, 0, 3, -2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 5, 2, 0, 0, 0, -2, -12, 0, 0, -4, 0, 0, 0, 0, 0, -6, 0, 2, 0, 0, 0, -4, 0, 7, 3
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Dirichlet convolution of this sequence with A348976 is A349338.
The positions of records start as: 1, 12, 18, 36, 100, 108, 196, 225, 324, 441, 500, 1125, 1372, 2500, 5000, 5324, 8575, 8788, 9604, 12500, 19652, etc.

Links

Crossrefs

Cf. A003415, A129283, A230593, A347084, A349434 (Dirichlet inverse), A349436 (sum with it).
Cf. also A348976, A349338.

Programs

  • Mathematica
    s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); dinv[1] = 1; dinv[n_] := dinv[n] = -DivisorSum[n, dinv[#] * d[n/#] &, # < n &]; a[n_] := DivisorSum[n, s[#] * dinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    up_to = 20000;
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]));
A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
A347084(n) = v347084[n];
A349435(n) = sumdiv(n,d,A230593(n/d)*A347084(d));

Formula

a(n) = Sum_{d|n} A230593(n/d) * A347084(d).
Showing 1-4 of 4 results.

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