A390554 a(n) is the sum of the bi-unitary divisors of n^2.

1, 5, 10, 27, 26, 50, 50, 119, 112, 130, 122, 270, 170, 250, 260, 495, 290, 560, 362, 702, 500, 610, 530, 1190, 756, 850, 1066, 1350, 842, 1300, 962, 2015, 1220, 1450, 1300, 3024, 1370, 1810, 1700, 3094, 1682, 2500, 1850, 3294, 2912, 2650, 2210, 4950, 2752, 3780

Offset

1,2

Comments

The number of bi-unitary divisors of n^2 is A322327(n).

Links

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

Formula

a(n) = A188999(n^2).

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

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 1.38365632653061275303... .

a(n) = sigma(n / rad(n)) * usigma(n * rad(n)), where sigma = A000203, usigma = A034448, rad = A007947. - Aloe Poliszuk, Nov 11 2025

Mathematica

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

Prog

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

Crossrefs

Cf. A000203, A000290, A007947, A034448, A188999, A322327.

Cf. A002117, A013661, A183699.

Similar sequences: A001157 (sum of square divisors of n^2), A034676, A065764, A374539, A380322, A390556.

Sequence in context: A128665 A218466 A277825 * A054298 A391588 A022094

Adjacent sequences: A390551 A390552 A390553 * A390555 A390556 A390557

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 10 2025

Status

approved