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 1-3 of 3 results.

A349623 Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

1, -1, -2, -10, -1, 2, -2, 18, -25, 1, -5, 20, -4, 2, 2, 46, -3, 25, -2, 10, 4, 5, -6, -36, -33, 4, 86, 20, -1, -2, -17, -220, 10, 3, 2, 250, -10, 2, 8, -18, -7, -4, -2, 50, 25, 6, -8, -92, -81, 33, 6, 40, -6, -86, 5, -36, 4, 1, -29, -20, -13, 17, 50, -886, 4, -10, -4, 30, 12, -2, -31, -450, -3, 10, 66, 20, 10, -8, -10
Offset: 1

Views

Author

Antti Karttunen, Nov 26 2021

Keywords

Comments

Multiplicative because A326042 is.

Links

Crossrefs

Cf. A000203, A003961, A064989, A326042, A349624, A349625, A349626.

Programs

  • Mathematica
    f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
memoA349623 = Map();
A349623(n) = if(1==n,1,my(v); if(mapisdefined(memoA349623,n,&v), v, v = -sumdiv(n,d,if(dA326042(n/d)*A349623(d),0)); mapput(memoA349623,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A326042(n/d) * a(d).

A349625 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

1, 1, 1, -8, 4, 1, 5, 2, -22, 4, 6, -8, 9, 5, 4, 50, 14, -22, 17, -32, 5, 6, 17, 2, -13, 9, 20, -40, 28, 4, 14, -120, 6, 14, 20, 176, 27, 17, 9, 8, 34, 5, 41, -48, -88, 17, 39, 50, -46, -13, 14, -72, 47, 20, 24, 10, 17, 28, 30, -32, 48, 14, -110, -1126, 36, 6, 63, -112, 17, 20, 40, -44, 70, 27, -13, -136, 30, 9, 69
Offset: 1

Views

Author

Antti Karttunen, Nov 26 2021

Keywords

Comments

Multiplicative because A000027 and A349623 are.

Links

Crossrefs

Cf. A000203, A003961, A064989, A326042, A349623, A349624 (Dirichlet inverse).

Programs

  • Mathematica
    f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
memoA349623 = Map();
A349623(n) = if(1==n,1,my(v); if(mapisdefined(memoA349623,n,&v), v, v = -sumdiv(n,d,if(dA326042(n/d)*A349623(d),0)); mapput(memoA349623,n,v); (v)));
A349625(n) = sumdiv(n,d,d*A349623(n/d));

Formula

a(n) = Sum_{d|n} d * A349623(n/d).

A349626 Möbius transform of A326042, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

1, 0, 1, 10, 0, 0, 1, -8, 27, 0, 4, 10, 3, 0, 0, 46, 2, 0, 1, 0, 1, 0, 5, -8, 33, 0, -7, 10, 0, 0, 16, 6, 4, 0, 0, 270, 9, 0, 3, 0, 6, 0, 1, 40, 0, 0, 7, 46, 83, 0, 2, 30, 5, 0, 0, -8, 1, 0, 28, 0, 12, 0, 27, 1036, 0, 0, 3, 20, 5, 0, 30, -216, 2, 0, 33, 10, 4, 0, 9, 0, 447, 0, 11, 10, 0, 0, 0, -32, 24, 0, 3, 50, 16
Offset: 1

Views

Author

Antti Karttunen, Nov 26 2021

Keywords

Comments

Dirichlet convolution of Euler phi (A000010) with A349624.
Multiplicative because A326042 is.

Links

Crossrefs

Cf. A000010, A003961, A008683, A064989, A326042, A349623, A349624.

Programs

  • Mathematica
    f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; a[n_] := DivisorSum[n, s[#] * MoebiusMu[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A349626(n) = sumdiv(n,d,moebius(n/d)*A326042(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A326042(d).
a(n) = Sum_{d|n} A000010(n/d) * A349624(d).
Showing 1-3 of 3 results.

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