A390556 The sum of the exponential divisors of n^2.

1, 6, 12, 22, 30, 72, 56, 78, 93, 180, 132, 264, 182, 336, 360, 278, 306, 558, 380, 660, 672, 792, 552, 936, 655, 1092, 768, 1232, 870, 2160, 992, 1062, 1584, 1836, 1680, 2046, 1406, 2280, 2184, 2340, 1722, 4032, 1892, 2904, 2790, 3312, 2256, 3336, 2457, 3930, 3672

Offset

1,2

Comments

The number of exponential divisors of n^2 is A390555(n).

Links

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

Formula

a(n) = A051377(n^2).

Multiplicative with a(p^e) = Sum_{d | (2*e)} p^d.

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} p^(2*k-1)/(p^(6*k-3) - 1) + Sum_{k>=1} p^(2*k)/(p^(3*k) - 1))) = 1.39919293179391004774... .

Mathematica

f[p_, e_] := DivisorSum[2*e, p^# &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

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

Crossrefs

Cf. A000290, A051377, A322791, A390555.

Similar sequences: A001157 (sum of square divisors of n^2), A034676, A065764, A374539, A380322, A390554.

Sequence in context: A360570 A178733 A344033 * A266085 A144568 A222001

Adjacent sequences: A390553 A390554 A390555 * A390557 A390558 A390559

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 10 2025

Status

approved