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 1-5 of 5 results.

A347130 a(n) = Sum_{d|n} d * A003415(n/d), where A003415 is the arithmetic derivative.

Original entry on oeis.org

0, 1, 1, 6, 1, 10, 1, 24, 9, 14, 1, 48, 1, 18, 16, 80, 1, 63, 1, 72, 20, 26, 1, 176, 15, 30, 54, 96, 1, 124, 1, 240, 28, 38, 24, 270, 1, 42, 32, 272, 1, 164, 1, 144, 117, 50, 1, 560, 21, 135, 40, 168, 1, 324, 32, 368, 44, 62, 1, 552, 1, 66, 153, 672, 36, 244, 1, 216, 52, 236, 1, 936, 1, 78, 165, 240, 36, 284, 1, 880
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Comments

Dirichlet convolution of the identity function (A000027) with the arithmetic derivative of n (A003415).
Dirichlet convolution of Euler phi (A000010) with A319684.

Links

Crossrefs

Inverse Möbius transform of A347131.
Cf. A000010, A000027, A003415, A003557, A319684, A347129.
Cf. also A300251, A305809, A319684, A347132, A347133.

Programs

  • Mathematica
    Table[DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &], {n, 80}] (* Michael De Vlieger, Oct 21 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));

Formula

a(n) = Sum_{d|n} d * A003415(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A319684(d).
a(n) = Sum_{d|n} A347131(d).
a(n) = A003557(n) * A347129(n).

A347131 a(n) = Sum_{d|n} phi(n/d) * A003415(d), where A003415 is the arithmetic derivative and phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 5, 1, 8, 1, 18, 8, 12, 1, 33, 1, 16, 14, 56, 1, 45, 1, 53, 18, 24, 1, 110, 14, 28, 45, 73, 1, 87, 1, 160, 26, 36, 22, 169, 1, 40, 30, 182, 1, 119, 1, 113, 93, 48, 1, 328, 20, 107, 38, 133, 1, 216, 30, 254, 42, 60, 1, 337, 1, 64, 125, 432, 34, 183, 1, 173, 50, 183, 1, 538, 1, 76, 135, 193, 34, 215, 1, 552, 216
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Comments

Dirichlet convolution of A000010 with A003415.

Links

Crossrefs

Möbius transform of A347130.
Cf. A000010, A003415.
Cf. also A300251, A305809, A322577, A347132, A347133, A347134, A347235.

Programs

  • Mathematica
    f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Sep 03 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347131(n) = sumdiv(n,d,A003415(n/d)*eulerphi(d));
  • PARI
    A347131(n) = sum(k=1,n,A003415(gcd(n,k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

Formula

a(n) = Sum_{d|n} A000010(n/d) * A003415(d).
a(n) = Sum_{d|n} A008683(n/d) * A347130(d).
a(n) = Sum_{k=1..n} A003415(gcd(n,k)). - Antti Karttunen, Sep 02 2021

A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

Original entry on oeis.org

0, 1, 1, 7, 1, 12, 1, 30, 10, 16, 1, 65, 1, 20, 18, 104, 1, 83, 1, 93, 22, 28, 1, 254, 16, 32, 63, 121, 1, 167, 1, 320, 30, 40, 26, 391, 1, 44, 34, 374, 1, 215, 1, 177, 143, 52, 1, 840, 22, 165, 42, 205, 1, 450, 34, 494, 46, 64, 1, 827, 1, 68, 183, 912, 38, 311, 1, 261, 54, 295, 1, 1430, 1, 80, 197, 289, 38, 359
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Comments

Dirichlet convolution of Dedekind psi function (A001615) with the arithmetic derivative (A003415).

Links

Crossrefs

Cf. A001615, A003415.
Cf. also A305809, A322577, A347131, A347133, A347135.

Programs

  • Mathematica
    Table[DivisorSum[n, DirichletConvolve[j, MoebiusMu[j]^2, j, n/#]*If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &], {n, 78}] (* Michael De Vlieger, Oct 19 2021, after Jan Mangaldan at A001615 *)
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347132(n) = sumdiv(n,d,A001615(n/d)*A003415(d));

Formula

a(n) = Sum_{d|n} A001615(n/d) * A003415(d).

A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

Original entry on oeis.org

0, 1, 1, 5, 1, 12, 1, 16, 7, 16, 1, 51, 1, 20, 18, 44, 1, 68, 1, 71, 22, 28, 1, 156, 11, 32, 33, 91, 1, 167, 1, 112, 30, 40, 26, 277, 1, 44, 34, 220, 1, 215, 1, 131, 110, 52, 1, 420, 15, 140, 42, 151, 1, 300, 34, 284, 46, 64, 1, 673, 1, 68, 138, 272, 38, 311, 1, 191, 54, 295, 1, 836, 1, 80, 162, 211, 38, 359, 1, 596
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Comments

Dirichlet convolution of A001615 (Dedekind psi function) with A069359.
Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A322577.

Links

Crossrefs

Cf. A001221, A001615, A069359, A322577.
Cf. also A347132, A347133, A347134.

Programs

  • Mathematica
    Table[DivisorSum[n,PrimeNu[n/#]*Sum[DirichletConvolve[j,MoebiusMu[j]^2,j,#/d] EulerPhi[d],{d,Divisors[#]}]&],{n,80}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347135(n) = sumdiv(n,d,A001615(n/d)*A069359(d));

Formula

a(n) = Sum_{d|n} A001615(n/d) * A069359(d).
a(n) = Sum_{d|n} A001221(n/d) * A322577(d).

A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 8, 5, 12, 1, 23, 1, 16, 14, 20, 1, 36, 1, 35, 18, 24, 1, 60, 9, 28, 21, 47, 1, 87, 1, 48, 26, 36, 22, 103, 1, 40, 30, 92, 1, 119, 1, 71, 66, 48, 1, 148, 13, 92, 38, 83, 1, 144, 30, 124, 42, 60, 1, 247, 1, 64, 86, 112, 34, 183, 1, 107, 50, 183, 1, 268, 1, 76, 110, 119, 34, 215, 1, 228, 81, 84
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Comments

Dirichlet convolution of A000010 (Euler totient function phi) with A069359.
Dirichlet convolution of A001221 (omega) with A029935 (the convolution square of Euler phi).

Links

Crossrefs

Cf. A000010, A001221, A069359, A029935.
Cf. also A347131, A347133, A347135.

Programs

Formula

a(n) = Sum_{d|n} A000010(n/d) * A069359(d)
a(n) = Sum_{d|n} A001221(n/d) * A029935(d).
a(n) = Sum_{k=1..n} A069359(gcd(n,k)). - Antti Karttunen, Oct 17 2021
Showing 1-5 of 5 results.

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