A369757 The number of divisors of the smallest cubefull exponentially odd number that is divisible by n.

1, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 16, 4, 16, 16, 6, 4, 16, 4, 16, 16, 16, 4, 16, 4, 16, 4, 16, 4, 64, 4, 6, 16, 16, 16, 16, 4, 16, 16, 16, 4, 64, 4, 16, 16, 16, 4, 24, 4, 16, 16, 16, 4, 16, 16, 16, 16, 16, 4, 64, 4, 16, 16, 8, 16, 64, 4, 16, 16, 64, 4, 16, 4

Offset

1,2

Links

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

Formula

a(n) = A000005(A356192(n)).

Multiplicative with a(p^e) = max(e,3) + 1 if e is odd, and e+2 if e is even.

a(n) >= A000005(n), with equality if and only if n is cubefull exponentially odd number (A335988).

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

From Vaclav Kotesovec, Feb 02 2024: (Start)

Dirichlet g.f.: zeta(s)^4 * Product_{p prime} (1 + (7*p^(2*s) + 2*p^(3*s) - 6*p^(4*s) - 7*p^s + 2) / ((p^s+1)*p^(5*s))).

Sum_{k=1..n} a(k) = c * n*log(n)^3/6 + O(n*log(n)^2), where c = Product_{p prime} (1 - (6*p^4 - 2*p^3 - 7*p^2 + 7*p - 2) / ((p+1)*p^5)) = 0.124604542136592401049820049658828040278... (End)

Mathematica

f[p_, e_] := If[OddQ[e], Max[e, 3] + 1, e + 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> if(x%2, max(x, 3) + 1, x + 2), factor(n)[, 2]));

Crossrefs

Cf. A000005, A335988, A356192, A365348, A369758, A369759.

Sequence in context: A358509 A285052 A369719 * A365489 A202686 A134660

Adjacent sequences: A369754 A369755 A369756 * A369758 A369759 A369760

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 31 2024

Status

approved