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.

A349443 a(n) = A349441(n) + A349442(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, -3, 4, 8, 0, -6, 0, 12, 16, -1, 0, -8, 0, -12, 24, 20, 0, -6, 16, 24, -16, -18, 0, 0, 0, 5, 40, 32, 48, 14, 0, 36, 48, -12, 0, 0, 0, -30, -32, 44, 0, -2, 36, -24, 64, -36, 0, -16, 80, -18, 72, 56, 0, 8, 0, 60, -48, 7, 96, 0, 0, -48, 88, 0, 0, 6, 0, 72, -48, -54, 120, 0, 0, -4, -20, 80, 0, 12
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Links

Crossrefs

Cf. A349441, A349442.

Programs

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349441(d) * A349442(n/d). [As the sequences are Dirichlet inverses of each other]

A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, -1, -2, 2, -4, 2, -6, 0, 6, 4, -10, -4, -12, 6, 8, 0, -16, -6, -18, -8, 12, 10, -22, 0, 20, 12, 0, -12, -28, -8, -30, 0, 20, 16, 24, 12, -36, 18, 24, 0, -40, -12, -42, -20, -24, 22, -46, 0, 42, -20, 32, -24, -52, 0, 40, 0, 36, 28, -58, 16, -60, 30, -36, 0, 48, -20, -66, -32, 44, -24, -70, 0, -72, 36, -40, -36
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Comments

Multiplicative because A055615 and A057521 are.
Convolving this with Euler phi (A000010) produces A349379.

Links

Crossrefs

Cf. A055615, A057521, A349442 (Dirichlet inverse), A349443 (sum with it).
Cf. also A097945, A349379.

Programs

  • Mathematica
    f[p_, e_] := Which[e > 2, 0, e == 2, p^2 - p, e == 1, 1 - p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)
  • PARI
    A055615(n) = (n*moebius(n));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349441(n) = sumdiv(n,d,A057521(n/d)*A055615(d));

Formula

a(n) = Sum_{d|n} A057521(n/d) * A055615(d).
Multiplicative with a(p^e) = 1 - p is e = 1, p^2 - p if e = 2, and 0 otherwise. - Amiram Eldar, Nov 19 2021

A349350 Dirichlet inverse of A057521, the powerful part of n.

Original entry on oeis.org

1, -1, -1, -3, -1, 1, -1, -1, -8, 1, -1, 3, -1, 1, 1, 5, -1, 8, -1, 3, 1, 1, -1, 1, -24, 1, -10, 3, -1, -1, -1, 7, 1, 1, 1, 24, -1, 1, 1, 1, -1, -1, -1, 3, 8, 1, -1, -5, -48, 24, 1, 3, -1, 10, 1, 1, 1, 1, -1, -3, -1, 1, 8, -3, 1, -1, -1, 3, 1, -1, -1, 8, -1, 1, 24, 3, 1, -1, -1, -5, 28, 1, -1, -3, 1, 1, 1, 1, -1, -8
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Comments

Multiplicative because A057521 is.

Links

Crossrefs

Cf. A057521.
Cf. also A349340, A349442.

Programs

  • Mathematica
    f[p_, e_] := Module[{B = 1 + p - 2*p^2, C = Sqrt[1 + 2*p - 3*p^2]}, FullSimplify[((B - C)*(p - 1 + C)^(e - 1) - (B + C)*(p - 1 - C)^(e - 1))/(2^e*C)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2023 *)
  • 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)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A057521(n/d) * a(d).
Let p be a prime, B = 1 + p - 2*p^2 and C = sqrt(1 + 2*p - 3*p^2). Then the sequence is multiplicative with a(p^e) = ((B-C)*(p-1+C)^(e-1) - (B+C)*(p-1-C)^(e-1))/(2^e*C). - Sebastian Karlsson, Dec 02 2021
Showing 1-3 of 3 results.

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