A369721 The sum of unitary divisors of the smallest cubefull number that is a multiple of n.

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 17, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 33, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512

Offset: 1

Author

Amiram Eldar, Jan 30 2024

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A034448, A036966, A356193, A369718, A369719, A369720.

Cf. A002117, A013662, A183700.

Programs

Mathematica

f[p_, e_] := If[e <= 2, p^3 + 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= 2, 1 + f[i,1]^3, 1 + f[i,1]^f[i,2]));}

Formula

a(n) = A034448(A356193(n)).

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

a(n) >= A034448(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-1) - 1/p^(2*s-4) + 1/p^(4*s-4) - 1/p^(4*s-3) ).

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^12 - 2/p^13 + 1/p^14) = 0.65803546696642353777... .