A358319 Multiplicative sequence a(n) with a(p^e) = ((p-2) - (p-1) * e) * p^(e-1) for prime p and e > 0.

1, -1, -1, -4, -1, 1, -1, -12, -9, 1, -1, 4, -1, 1, 1, -32, -1, 9, -1, 4, 1, 1, -1, 12, -25, 1, -45, 4, -1, -1, -1, -80, 1, 1, 1, 36, -1, 1, 1, 12, -1, -1, -1, 4, 9, 1, -1, 32, -49, 25, 1, 4, -1, 45, 1, 12, 1, 1, -1, -4, -1, 1, 9, -192, 1, -1, -1, 4, 1, -1, -1, 108, -1, 1, 25, 4, 1, -1, -1, 32

Offset

1,4

Links

Table of n, a(n) for n=1..80.

Formula

Equals Dirichlet convolution of A000010 and n * A076479.

Dirichlet g.f.: (zeta(s-1)^2 / zeta(s)) * Product_{primes p} (1 - 2 / p^(s-1)).

Dirichlet inverse b(n), n > 0, is multiplicative with b(p^e) = (p^2 * (2*p)^(e-1) - (p-1)^2) / (2*p-1) for prime p and e > 0.

Conjecture: a(n) = Sum_{k=1..n} gcd(k, n) * A076479(gcd(k, n)) for n > 0.

Mathematica

f[p_, e_]: = ((p-2) - (p-1) * e) * p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2022 *)

Prog

(PARI) a(n, f=factor(n))=prod(i=1, #f~, ((f[i, 1]-2) - (f[i, 1]-1) * f[i, 2]) * f[i, 1]^(f[i, 2]-1)) \\ Charles R Greathouse IV, Nov 09 2022

Crossrefs

Cf. A000010, A076479.

Sequence in context: A124777 A352834 A203639 * A265679 A348989 A362297

Adjacent sequences: A358316 A358317 A358318 * A358320 A358321 A358322

Keyword

sign,easy,mult

Author

Werner Schulte, Nov 09 2022

Status

approved