A368470 a(n) is the number of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset

1,2

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Formula

a(n) = A033634(A350389(n)).

Multiplicative with a(p^e) = (e+3)/2 if e is odd and 1 otherwise.

a(n) >= 1, with equality if and only if n is a square (A000290).

a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).

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

From Vaclav Kotesovec, Dec 26 2023: (Start)

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

Let f(s) = Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).

Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - (2*p+1) / (p*(p+1)^2)) = 0.528940778823659679133966695786017426052491935740673837882972347697...,

f'(1) = f(1) * Sum_{p prime} (4*p^2 + 3*p + 1) * log(p) / (p^4 + 3*p^3 + p^2 - 2*p - 1) = f(1) * 1.36109933267802415215189866467122940932493907539386280428818...

and gamma is the Euler-Mascheroni constant A001620. (End)

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A000290, A005117, A033634, A268335, A350389, A368471.

Sequence in context: A144925 A029262 A363825 * A295878 A368977 A294931

Adjacent sequences: A368467 A368468 A368469 * A368471 A368472 A368473

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 26 2023

Status

approved