A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

1, 3, 4, 3, 6, 12, 8, 5, 4, 18, 12, 12, 14, 24, 24, 5, 18, 12, 20, 18, 32, 36, 24, 20, 6, 42, 10, 24, 30, 72, 32, 9, 48, 54, 48, 12, 38, 60, 56, 30, 42, 96, 44, 36, 24, 72, 48, 20, 8, 18, 72, 42, 54, 30, 72, 40, 80, 90, 60, 72, 62, 96, 32, 9, 84, 144, 68, 54, 96

Offset

1,2

Comments

The number of unitary divisors of the smallest number whose square is 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(A019554(n)).

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

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

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6) = 0.515959523197... .

Mathematica

f[p_, e_] := p^Ceiling[e/2] + 1; 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]^ceil(f[i, 2]/2) + 1); }
(Python)
from math import prod
from sympy import factorint
def A365481(n): return prod(p**((e>>1)+(e&1))+1 for p, e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

Crossrefs

Cf. A002117, A013661, A019554, A034444, A034448, A365347, A365479, A365480.

Sequence in context: A046897 A109506 A369889 * A000113 A365347 A069915

Adjacent sequences: A365478 A365479 A365480 * A365482 A365483 A365484

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 05 2023

Status

approved