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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.

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A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).

Original entry on oeis.org

1, 1, 2, -1, 4, 2, 6, -3, -2, 4, 10, -2, 12, 6, 8, -1, 16, -2, 18, -4, 12, 10, 22, -6, -4, 12, -16, -6, 28, 8, 30, 5, 20, 16, 24, 2, 36, 18, 24, -12, 40, 12, 42, -10, -8, 22, 46, -2, -6, -4, 32, -12, 52, -16, 40, -18, 36, 28, 58, -8, 60, 30, -12, 7, 48, 20, 66, -16, 44, 24, 70, 6, 72, 36, -8, -18, 60, 24, 78, -4
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Comments

Multiplicative because both A000027 and A349350 are.

Links

Crossrefs

Cf. A000027, A057521, A349350, A349441 (Dirichlet inverse), A349443 (sum with it).

Programs

  • PARI
    A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
memoA349350 = Map();
A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(dA057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));
A349442(n) = sumdiv(n,d,d*A349350(n/d));

Formula

a(n) = Sum_{d|n} d * A349350(n/d).

A349340 Dirichlet inverse of A003557, where A003557 is multiplicative with a(p^e) = p^(e-1).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, -1, -2, 1, -1, 1, -1, 1, 1, -1, -1, 2, -1, 1, 1, 1, -1, 1, -4, 1, -4, 1, -1, -1, -1, -1, 1, 1, 1, 2, -1, 1, 1, 1, -1, -1, -1, 1, 2, 1, -1, 1, -6, 4, 1, 1, -1, 4, 1, 1, 1, 1, -1, -1, -1, 1, 2, -1, 1, -1, -1, 1, 1, -1, -1, 2, -1, 1, 4, 1, 1, -1, -1, 1, -8, 1, -1, -1, 1, 1, 1, 1, -1, -2, 1
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Links

Crossrefs

Cf. A003557, A076479, A326297 (absolute values).
Cf. also A325126, A349350, A349619.

Programs

  • Mathematica
    f[p_, e_] := -(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    A003557(n) = (n/factorback(factorint(n)[, 1]));
memoA349340 = Map();
A349340(n) = if(1==n,1,my(v); if(mapisdefined(memoA349340,n,&v), v, v = -sumdiv(n,d,if(dA003557(n/d)*A349340(d),0)); mapput(memoA349340,n,v); (v)));
  • PARI
    A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };

Formula

Multiplicative with a(p^e) = -((p-1)^(e-1)).
a(n) = A076479(n) * A326297(n).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A003557(n/d) * a(d).
Showing 1-2 of 2 results.

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