A358346 - OEIS

%I #19 Jul 09 2024 00:53:51

%S 1,3,4,1,6,12,8,9,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,36,1,42,28,

%T 8,30,72,32,33,48,54,48,1,38,60,56,54,42,96,44,12,6,72,48,4,1,3,72,14,

%U 54,84,72,72,80,90,60,24,62,96,8,1,84,144,68,18,96,144

%N a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).

%C The number of unitary divisors of n that are exponentially odd is A055076(n).

%H Amiram Eldar, <a href="/A358346/b358346.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) >= 1 with equality if and only if n is a square (A000290).

%F a(n) <= A033634(n) with equality if and only if n is squarefree (A005117).

%F Multiplicative with a(p^e) = p^e + 1 if e is odd, and 1 otherwise.

%F a(n) = A034448(n)/A358347(n).

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

%F From _Amiram Eldar_, Sep 14 2023: (Start)

%F a(n) = A034448(A350389(n)).

%F Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(2*s-1)). (End)

%t f[p_, e_] := 1 + If[OddQ[e], p^e, 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i,2]%2,  f[i,1]^f[i,2], 0));}

%Y Cf. A000290, A005117, A055076, A077610, A268335, A351569.

%Y Similar sequences: A033634, A034448, A035316, A358347.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Nov 11 2022