A346613 Inverse Moebius transform of A019554.

1, 3, 4, 5, 6, 12, 8, 9, 7, 18, 12, 20, 14, 24, 24, 13, 18, 21, 20, 30, 32, 36, 24, 36, 11, 42, 16, 40, 30, 72, 32, 21, 48, 54, 48, 35, 38, 60, 56, 54, 42, 96, 44, 60, 42, 72, 48, 52, 15, 33, 72, 70, 54, 48, 72, 72, 80, 90, 60, 120, 62, 96, 56, 29, 84, 144, 68, 90, 96, 144, 72, 63

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Transforms.

Formula

From Amiram Eldar, Oct 30 2025: (Start)

Multiplicative with a(p^e) = 2*(p^((e+1)/2+1)-1)/(p-1) - p^((e+1)/2) - 1 if e is odd, and 2*(p^(e/2+1)-1)/(p-1) - 1 if e is even.

Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) / zeta(2*s-2).

Sum_{k=1..n} a(k) ~ zeta(3) * n^2 / 2. (End)

Maple

A019554:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(t[1]^(ceil(t[2]/2)), t=F)
end proc:
f:= proc(n) local d; add(A019554(d), d = numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Oct 30 2025

Mathematica

f[p_, e_] := 2*(p^(Ceiling[e/2] + 1) - 1)/(p - 1) - 1 - If[OddQ[e], p^Ceiling[e/2], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2025 *)

Prog

(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; 2*(p^(ceil(e/2) + 1) - 1)/(p - 1) - 1 - if(e % 2, p^ceil(e/2))); } \\ Amiram Eldar, Oct 30 2025

Crossrefs

Cf. A002117, A019554, A346612.

Sequence in context: A191750 A384554 A394255 * A378433 A378434 A385049

Adjacent sequences: A346610 A346611 A346612 * A346614 A346615 A346616

Keyword

nonn,mult,easy

Author

N. J. A. Sloane, Aug 18 2021

Status

approved