Amiram Eldar, <a href="/A369720/b369720_1.txt">Table of n, a(n) for n = 1..10000</a>

reviewed

approved

proposed

reviewed

editing

proposed

Amiram Eldar, <a href="/A369720/b369720_1.txt">Table of n, a(n) for n = 1..10000</a>

allocated for Amiram EldarThe sum of divisors of the smallest cubefull number that is a multiple of n.

1, 15, 40, 15, 156, 600, 400, 15, 40, 2340, 1464, 600, 2380, 6000, 6240, 31, 5220, 600, 7240, 2340, 16000, 21960, 12720, 600, 156, 35700, 40, 6000, 25260, 93600, 30784, 63, 58560, 78300, 62400, 600, 52060, 108600, 95200, 2340, 70644, 240000, 81400, 21960, 6240

1,2

a(n) = A000203(A356193(n)).

Multiplicative with a(p) = p^3 + p^2 + p + 1 for e <= 2, and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 3.

a(n) >= A000203(n), with equality if and only if n is cubefull (A036966).

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(4*s-4)).

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^3 - 1/p^4 + 1/p^7 + 1/p^12 - 1/p^13) = 1.00015013207437782094... .

f[p_, e_] := (p^If[e <= 2, 4, e + 1]-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

(PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] <= 2, f[i, 2] = 3)); sigma(f); }

Cf. A000203, A036966, A356193, A369717, A369719, A369721.

Cf. A002117, A013662, A183700.

allocated

nonn,easy,mult

Amiram Eldar, Jan 30 2024

approved

editing

allocated for Amiram Eldar

allocated

approved