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.

A322577 a(n) = Sum_{d|n} psi(n/d) * phi(d).

Original entry on oeis.org

1, 4, 6, 11, 10, 24, 14, 28, 26, 40, 22, 66, 26, 56, 60, 68, 34, 104, 38, 110, 84, 88, 46, 168, 74, 104, 102, 154, 58, 240, 62, 160, 132, 136, 140, 286, 74, 152, 156, 280, 82, 336, 86, 242, 260, 184, 94, 408, 146, 296, 204, 286, 106, 408, 220, 392, 228, 232, 118, 660
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 29 2019

Keywords

Comments

Dirichlet convolution of Dedekind psi function (A001615) with Euler totient function (A000010).
Dirichlet convolution of A008966 with A018804.
Dirichlet convolution of A038040 with A271102.

Links

Crossrefs

Cf. A000010, A001615, A008966, A018804, A038040, A271102.
Cf. A327251 (inverse Möbius transform), A347092 (Dirichlet inverse), A347093 (sum with it), A347135.

Programs

  • Maple
    f:= proc(n) local t;
  mul((t[2]+1)*t[1]^t[2] - (t[2]-1)*t[1]^(t[2]-2), t = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2019
  • Mathematica
    Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e + 1)*p^e - (e - 1)*p^(e - 2); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)
  • PARI
    seq(n) = {dirmul(vector(n, n, eulerphi(n)), vector(n, n, n * sumdivmult(n, d, issquarefree(d)/d)))} \\ Andrew Howroyd, Aug 29 2019
  • 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)));
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d)); \\ Antti Karttunen, Apr 03 2022

Formula

Dirichlet g.f.: zeta(s-1)^2 / zeta(2*s).
a(p) = 2*p, where p is prime.
Sum_{k=1..n} a(k) ~ 45*n^2*(2*Pi^4*log(n) - Pi^4 + 4*gamma*Pi^4 - 360*zeta'(4)) / (2*Pi^8), where gamma is the Euler-Mascheroni constant A001620 and for zeta'(4) see A261506. - Vaclav Kotesovec, Aug 31 2019
a(p^k) = (k+1)*p^k - (k-1)*p^(k-2) where p is prime. - Robert Israel, Sep 01 2019
a(n) = Sum_{k=1..n} psi(gcd(n,k)). - Ridouane Oudra, Nov 29 2019
a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A347094 Sum of A038040 (convolution of sigma with Euler phi) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 16, 0, 48, 0, 32, 36, 80, 0, 48, 0, 112, 120, 80, 0, 72, 0, 80, 168, 176, 0, 192, 100, 208, 108, 112, 0, 0, 0, 192, 264, 272, 280, 360, 0, 304, 312, 320, 0, 0, 0, 176, 180, 368, 0, 480, 196, 200, 408, 208, 0, 432, 440, 448, 456, 464, 0, 960, 0, 496, 252, 448, 520, 0, 0, 272, 552, 0, 0, 864, 0, 592, 300, 304, 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. A000005, A030059, A030229, A038040, A328722, A347093.

Programs

  • PARI
    up_to = 16384;
A038040(n) = (n*numdiv(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA038040(n)));
A328722(n) = v328722[n];
A347094(n) = (A038040(n)+A328722(n));

Formula

a(n) = A038040(n) + A328722(n).
For n > 1, a(n) = -Sum_{d|n, 1A038040(d) * A328722(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A038040(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)

A347091 Sum of A332844 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 21, 16, 36, 0, 28, 0, 48, 48, 37, 0, 36, 0, 42, 64, 72, 0, 60, 36, 84, 48, 56, 0, 0, 0, 81, 96, 108, 96, 114, 0, 120, 112, 90, 0, 0, 0, 84, 72, 144, 0, 164, 64, 84, 144, 98, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 96, 166, 168, 0, 0, 126, 192, 0, 0, 258, 0, 228, 112, 140, 192, 0, 0, 246, 132
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. A030059, A030229, A332844, A347090.
Cf. also A347093.

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(dA332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d));
v347090 = DirInverseCorrect(vector(up_to,n,A332844(n)));
A347090(n) = v347090[n];
A347091(n) = (A332844(n)+A347090(n));

Formula

a(n) = A332844(n) + A347090(n).
For n > 1, a(n) = -Sum_{d|n, 1A332844(d) * A347090(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A332844(A030229(n)).
Showing 1-4 of 4 results.

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