A358346 a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).

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

Offset

1,2

Comments

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

Links

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

Formula

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

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

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

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

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

From Amiram Eldar, Sep 14 2023: (Start)

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

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)

Mathematica

f[p_, e_] := 1 + If[OddQ[e], p^e, 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 11 2022 *)

Prog

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

Crossrefs

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

Similar sequences: A033634, A034448, A035316, A358347.

Sequence in context: A323159 A347090 A328181 * A348733 A368471 A351569

Adjacent sequences: A358343 A358344 A358345 * A358347 A358348 A358349

Keyword

nonn,easy,mult

Author

Amiram Eldar, Nov 11 2022

Status

approved