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 1-7 of 7 results.

A347093 Sum of A322577 (convolution of Dedekind psi with Euler phi) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 16, 0, 48, 0, 24, 36, 80, 0, 36, 0, 112, 120, 73, 0, 64, 0, 60, 168, 176, 0, 192, 100, 208, 96, 84, 0, 0, 0, 156, 264, 272, 280, 336, 0, 304, 312, 320, 0, 0, 0, 132, 160, 368, 0, 378, 196, 192, 408, 156, 0, 432, 440, 448, 456, 464, 0, 960, 0, 496, 224, 373, 520, 0, 0, 204, 552, 0, 0, 688, 0, 592, 288, 228, 616
Offset: 1

Views

Author

Antti Karttunen, Aug 18 2021

Keywords

Comments

No negative terms in range 1 .. 2^20.
Apparently, A030059 gives the positions of all zeros.

Links

Crossrefs

Cf. A000010, A001615, A030059, A030229, A322577, A347092, A347094.

Programs

  • PARI
    up_to = 16384;
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
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA322577(n)));
A347092(n) = v347092[n];
A347093(n) = (A322577(n)+A347092(n));

Formula

a(n) = A322577(n) + A347092(n).
For n > 1, a(n) = -Sum_{d|n, 1A322577(d) * A347092(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A322577(A030229(n)).

A347092 Dirichlet inverse of A322577, which is the convolution of Dedekind psi with Euler phi.

Original entry on oeis.org

1, -4, -6, 5, -10, 24, -14, -4, 10, 40, -22, -30, -26, 56, 60, 5, -34, -40, -38, -50, 84, 88, -46, 24, 26, 104, -6, -70, -58, -240, -62, -4, 132, 136, 140, 50, -74, 152, 156, 40, -82, -336, -86, -110, -100, 184, -94, -30, 50, -104, 204, -130, -106, 24, 220, 56, 228, 232, -118, 300, -122, 248, -140, 5, 260, -528, -134
Offset: 1

Views

Author

Antti Karttunen, Aug 18 2021

Keywords

Comments

Multiplicative because A322577 is.

Links

Crossrefs

Cf. A000010, A001615, A322577, A347093.
Cf. A023900, A323363.

Programs

  • Haskell
    import Math.NumberTheory.Primes
a n = product . map (\(p, e) -> if even e then 1 + unPrime p^2 else -2*unPrime p) . factorise $ n -- Sebastian Karlsson, Oct 29 2021
  • Mathematica
    f[p_, e_] := If[EvenQ[e], p^2 + 1, -2*p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)
  • PARI
    up_to = 16384;
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
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA322577(n)));
A347092(n) = v347092[n];
  • PARI
    A347092(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]%2, -2*f[i, 1], 1+(f[i, 1]^2))); }; \\ (after Sebastian Karlsson's multiplicative formula) - Antti Karttunen, Nov 11 2021
  • Python
    from sympy import factorint, prod
def f(p, e): return 1 + p**2 if e%2 == 0 else -2*p
def a(n):
    factors = factorint(n)
    return prod(f(p, factors[p]) for p in factors) # Sebastian Karlsson, Oct 29 2021

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A322577(n/d) * a(d).
a(n) = A347093(n) - A322577(n).
From Sebastian Karlsson, Oct 29 2021: (Start)
Dirichlet g.f.: zeta(2*s)/zeta(s-1)^2.
a(n) = Sum_{d|n} A323363(n/d)*A023900(d).
Multiplicative with a(p^e) = 1 + p^2 if e is even, -2*p if e is odd. (End)

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).

A327251 Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.

Original entry on oeis.org

1, 5, 7, 16, 11, 35, 15, 44, 33, 55, 23, 112, 27, 75, 77, 112, 35, 165, 39, 176, 105, 115, 47, 308, 85, 135, 135, 240, 59, 385, 63, 272, 161, 175, 165, 528, 75, 195, 189, 484, 83, 525, 87, 368, 363, 235, 95, 784, 161, 425, 245, 432, 107, 675, 253, 660, 273
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 15 2019

Keywords

Comments

Inverse Moebius transform of A322577.
Dirichlet convolution of A001615 with A000027.

Crossrefs

Cf. A000027, A001615, A018804, A060648, A322577, A347127.

Programs

  • Mathematica
    nmax = 57; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := p^(e - 1)*((p + 1)*e + p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)
  • PARI
    mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sumdiv(n, d, mypsi(n/d)*d); \\ Michel Marcus, Sep 15 2019

Formula

a(n) = Sum_{d|n} psi(n/d) * d.
a(p) = 2*p + 1, where p is prime.
Multiplicative with a(p^e) = p^(e-1)*((p+1)*e + p). - Antti Karttunen, Aug 24 2021

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).

A341772 a(n) = Sum_{d|n} phi(d) * J_2(n/d).

Original entry on oeis.org

1, 4, 10, 17, 28, 40, 54, 70, 94, 112, 130, 170, 180, 216, 280, 284, 304, 376, 378, 476, 540, 520, 550, 700, 716, 720, 858, 918, 868, 1120, 990, 1144, 1300, 1216, 1512, 1598, 1404, 1512, 1800, 1960, 1720, 2160, 1890, 2210, 2632, 2200, 2254, 2840, 2682, 2864, 3040, 3060, 2860, 3432, 3640
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Comments

Dirichlet convolution of Euler totient function phi (A000010) with Jordan function J_2 (A007434).

Links

Crossrefs

Cf. A000010, A000203, A001615, A002618, A007427, A007431, A007434, A008683, A023900, A029935, A064987, A069097, A321322, A322577, A338164.

Programs

  • Mathematica
    Jordan2[n_] := Sum[MoebiusMu[n/d] d^2, {d, Divisors[n]}]; a[n_] := Sum[EulerPhi[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 55}]
f[p_, e_] := p^(e-3)*(p-1)*(p^e*(p+1)^2-p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 31 2024 *)
  • PARI
    J2(n) = sumdiv(n, d, d^2 * moebius(n/d)); \\ A007434
a(n) = sumdiv(n, d, eulerphi(d) * J2(n/d)); \\ Michel Marcus, Feb 20 2021

Formula

Dirichlet g.f.: zeta(s-1) * zeta(s-2) / zeta(s)^2.
a(n) = Sum_{k=1..n} J_2(gcd(n,k)).
a(n) = Sum_{d|n} psi(d) * phi(d) * phi(n/d).
a(n) = Sum_{d|n} d * phi(d) * A029935(n/d).
a(n) = Sum_{d|n} d * sigma(d) * A007427(n/d).
a(n) = Sum_{d|n} d * A321322(n/d).
a(n) = Sum_{d|n} d * A023900(d) * A338164(n/d).
a(n) = Sum_{d|n} d^2 * A007431(n/d).
a(n) = Sum_{d|n} mu(n/d) * A069097(d).
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)^2). - Vaclav Kotesovec, Feb 20 2021
a(n) = Sum_{k=1..n} J_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = Sum_{1 <= i, j <= n} phi(gcd(i, j, n)). - Peter Bala, Jan 21 2024
Multiplicative with a(p^e) = p^(e-3)*(p-1)*(p^e*(p+1)^2-p). - Amiram Eldar, May 31 2024
Showing 1-7 of 7 results.

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