A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

1, 5, 10, 1, 26, 50, 50, 65, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 650, 1, 850, 730, 50, 842, 1300, 962, 1025, 1220, 1450, 1300, 1, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 3650, 3172

Offset

1,2

Comments

The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).

Links

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

Formula

a(n) = A034676(A350389(n)).

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

a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).

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

Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-2)).

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .

Mathematica

f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

Prog

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

Crossrefs

Cf. A000290, A005117, A034676, A055076, A268335, A350389, A358346, A374537.

Cf. A002117, A013661, A183699.

Sequence in context: A099731 A307716 A091306 * A073048 A102258 A361672

Adjacent sequences: A374535 A374536 A374537 * A374539 A374540 A374541

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jul 11 2024

Status

approved