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

Author

Amiram Eldar, Nov 11 2022

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

Similar sequences: A033634, A034448, A035316, A358347.

Programs

Mathematica

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

PARI

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

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)