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.

A349436 a(n) = A349434(n) + A349435(n).

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 12
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Links

Crossrefs

Cf. A349434, A349435.

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]); dinv[1] = 1; dinv[n_] := dinv[n] = -DivisorSum[n, dinv[#] * d[n/#] &, # < n &];  a[n_] := DivisorSum[n, dinv[#] * s[n/#] + sinv[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    A349436(n) = (A349434(n) + A349435(n)); \\ Needs also code from A349434 and A349435.

Formula

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A349434(d) * A349435(n/d). [As the sequences are Dirichlet inverses of each other]

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).

A349337 Dirichlet inverse of A230593.

Original entry on oeis.org

1, -3, -4, 3, -6, 13, -8, -3, 4, 19, -12, -16, -14, 25, 25, 3, -18, -17, -20, -22, 33, 37, -24, 19, 6, 43, -4, -28, -30, -87, -32, -3, 49, 55, 49, 33, -38, 61, 57, 25, -42, -113, -44, -40, -29, 73, -48, -22, 8, -25, 73, -46, -54, 21, 73, 31, 81, 91, -60, 125, -62, 97, -37, 3, 85, -165, -68, -58, 97, -163, -72, -52
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Comments

Coincides with A347084 on all squarefree numbers (A005117), but also on n=81, where a(81) = A347084(81) = 4. Question: Are there any other such numbers?

Links

Crossrefs

Cf. A005117, A069359, A230593.
Cf. also A347084, A349434, A349435.

Programs

  • 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(dA230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
A349337(n) = v349337[n];

Formula

For n > 1, a(n) = -Sum_{d|n, 1A230593(d) * A349337(n/d).
Showing 1-4 of 4 results.

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