A365479 The sum of unitary divisors of the smallest square divisible by n.

1, 5, 10, 5, 26, 50, 50, 17, 10, 130, 122, 50, 170, 250, 260, 17, 290, 50, 362, 130, 500, 610, 530, 170, 26, 850, 82, 250, 842, 1300, 962, 65, 1220, 1450, 1300, 50, 1370, 1810, 1700, 442, 1682, 2500, 1850, 610, 260, 2650, 2210, 170, 50, 130, 2900, 850, 2810, 410

Offset

1,2

Comments

The number of unitary divisors of the smallest square divisible by n is the same as the number of unitary divisors of n, A034444(n).

Links

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

Formula

a(n) = A034448(A053143(n)).

Multiplicative with a(p^e) = p^(e + (e mod 2)) + 1.

Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-2) - 1/p^(3*s-2)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * zeta(3) * Product_{p prime} (1 - 1/p^4 + 1/p^5 - 1/p^6) = 0.248414056414... .

Mathematica

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

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2] + f[i, 2]%2) + 1); }
(Python)
from math import prod
from sympy import factorint
def A365479(n): return prod(p**(e+(e&1))+1 for p, e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

Crossrefs

Cf. A002117, A034444, A034448, A053143, A365346, A365480, A365481.

Sequence in context: A328639 A351265 A369718 * A374537 A360560 A326724

Adjacent sequences: A365476 A365477 A365478 * A365480 A365481 A365482

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 05 2023

Status

approved