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

1,2

Links

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

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... .

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]

Prog

(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])); }

Crossrefs

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

Cf. A002117, A013662, A183700.

Sequence in context: A053825 A328640 A351266 * A369759 A033479 A254139

Adjacent sequences: A369718 A369719 A369720 * A369722 A369723 A369724

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 30 2024

Status

approved