A346613 - OEIS

%I #19 Oct 30 2025 19:41:15

%S 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,

%T 42,16,40,30,72,32,21,48,54,48,35,38,60,56,54,42,96,44,60,42,72,48,52,

%U 15,33,72,70,54,48,72,72,80,90,60,120,62,96,56,29,84,144,68,90,96,144,72,63

%N Inverse Moebius transform of A019554.

%H Amiram Eldar, <a href="/A346613/b346613.txt">Table of n, a(n) for n = 1..10000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%F From _Amiram Eldar_, Oct 30 2025: (Start)

%F 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.

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

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

%p A019554:= proc(n) local F,t;

%p   F:= ifactors(n)[2];

%p   mul(t[1]^(ceil(t[2]/2)),t=F)

%p end proc:

%p f:= proc(n) local d; add(A019554(d), d = numtheory:-divisors(n)) end proc:

%p map(f, [$1..100]); # _Robert Israel_, Oct 30 2025

%t 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 *)

%o (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

%Y Cf. A002117, A019554, A346612.

%K nonn,mult,easy

%O 1,2

%A _N. J. A. Sloane_, Aug 18 2021