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.

A349446 a(n) = A349444(n) + A349445(n).

Original entry on oeis.org

2, 0, 0, 1, 0, -2, 0, 1, 1, -4, 0, -1, 0, -6, 4, 1, 0, -5, 0, -2, 6, -10, 0, -1, 4, -12, 5, -3, 0, -4, 0, 1, 10, -16, 12, -2, 0, -18, 12, -2, 0, -6, 0, -5, 14, -22, 0, -1, 9, -16, 16, -6, 0, -13, 20, -3, 18, -28, 0, 0, 0, -30, 21, 1, 24, -10, 0, -8, 22, -12, 0, -2, 0, -36, 24, -9, 30, -12, 0, -2, 19, -40, 0, 0, 32
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Links

Crossrefs

Cf. A349444, A349445.
Cf. also A349433.

Programs

  • Mathematica
    s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#]*k[n/#] &, # < n &]; a[n_] := DivisorSum[n,  s[#]*k[n/#] + IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)
  • PARI
    A349446(n) = (A349444(n)+A349445(n)); \\ Needs also code from A349444 and A349445.

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349444(d) * A349445(n/d). [As the sequences are Dirichlet inverses of each other]

A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).

Original entry on oeis.org

1, -1, 1, 0, 2, -1, 3, 0, 3, -2, 5, 0, 6, -3, 4, 0, 8, -3, 9, 0, 6, -5, 11, 0, 10, -6, 9, 0, 14, -4, 15, 0, 10, -8, 12, 0, 18, -9, 12, 0, 20, -6, 21, 0, 12, -11, 23, 0, 21, -10, 16, 0, 26, -9, 20, 0, 18, -14, 29, 0, 30, -15, 18, 0, 24, -10, 33, 0, 22, -12, 35, 0, 36, -18, 20, 0, 30, -12, 39, 0, 27, -20, 41, 0, 32
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Links

Crossrefs

Cf. A001511, A003602, A008683, A092673, A349445 (Dirichlet inverse), A349446 (sum with it).
Cf. also A349431, A349447.

Programs

  • Mathematica
    s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, s[#]*k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)
  • PARI
    A003602(n) = (1+(n>>valuation(n,2)))/2;
A092673(n) = if(n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2))); \\ From A092673
A349444(n) = sumdiv(n,d,A003602(n/d)*A092673(d));

Formula

a(n) = Sum_{d|n} A003602(n/d) * A092673(d).

A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 2, 2, 5, 2, 6, 3, 0, 8, 8, 2, 9, 4, 0, 5, 11, 4, 6, 6, 4, 6, 14, 0, 15, 16, 0, 8, 0, 4, 18, 9, 0, 8, 20, 0, 21, 10, -2, 11, 23, 8, 12, 6, 0, 12, 26, 4, 0, 12, 0, 14, 29, 0, 30, 15, -3, 32, 0, 0, 33, 16, 0, 0, 35, 8, 36, 18, -4, 18, 0, 0, 39, 16, 8, 20, 41, 0, 0, 21, 0, 20, 44, -2, 0, 22, 0, 23
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Links

Crossrefs

Cf. A003602, A055615, A349134, A349431 (Dirichlet inverse), A349433 (sum with it).
Cf. also A349445, A349448.

Programs

  • Mathematica
    k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#] * k[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • 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(dA003602(n) = (1+(n>>valuation(n,2)))/2;
v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
A349134(n) = v349134[n];
A003602(n) = (1+(n>>valuation(n,2)))/2;
A055615(n) = (n*moebius(n));
A349432(n) = sumdiv(n,d,d*A349134(n/d));

A349448 Dirichlet convolution of A000265 (odd part of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 6, 0, 0, 0, 8, 0, 9, 0, 0, 0, 11, 0, 6, 0, 4, 0, 14, 0, 15, 0, 0, 0, 0, 0, 18, 0, 0, 0, 20, 0, 21, 0, -2, 0, 23, 0, 12, 0, 0, 0, 26, 0, 0, 0, 0, 0, 29, 0, 30, 0, -3, 0, 0, 0, 33, 0, 0, 0, 35, 0, 36, 0, -4, 0, 0, 0, 39, 0, 8, 0, 41, 0, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, 0, 0, 48, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 19 2021

Keywords

Links

Crossrefs

Cf. A000265, A003602, A349134, A349447 (Dirichlet inverse).
Cf. also A349432, A349445.

Programs

  • Mathematica
    k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#]*k[n/#] &, # < n &]; a[n_] := DivisorSum[n, # / 2^IntegerExponent[#, 2] * kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)
  • PARI
    A000265(n) = (n >> valuation(n, 2));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349448(n) = sumdiv(n,d,A000265(d)*A349134(n/d));

Formula

a(n) = Sum_{d|n} A000265(d) * A349134(n/d).
From Bernard Schott, Dec 18 2021: (Start)
If p is an odd prime, a(p) = (p-1)/2.
If n is even, a(n) = 0. (End)
Showing 1-4 of 4 results.

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