OFFSET
1,2
COMMENTS
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
G.f.: Sum_{i>=1} Sum_{j>=0} x^(i*3^j) / (1 - x^(i*3^j)).
a(n) = Sum_{d|n} A051064(d).
Sum_{k=1..n} a(k) ~ 3*n*(log(n)/2 - log(3)/4 - 1/2 + gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(p^e) = (e+1)*(e+2)/2 if p=3, and e+1 otherwise. - Amiram Eldar, Dec 02 2020
From Ridouane Oudra, Sep 30 2024: (Start)
a(n) = Sum_{i=0..A007949(n)} tau(n/3^i).
a(n) = Sum_{d|3*n} A007949(d).
MAPLE
seq(add(padic[ordp](3*d, 3), d in numtheory[divisors](n)), n=1..100); # Ridouane Oudra, Sep 30 2024
MATHEMATICA
Table[DivisorSum[n, IntegerExponent[3 #, 3] &], {n, 1, 85}]
nmax = 85; CoefficientList[Series[Sum[Sum[x^(i 3^j)/(1 - x^(i 3^j)), {j, 0, Floor[Log[3, nmax]] + 1}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := If[p == 3, (e + 1)*(e + 2)/2, e + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, Oct 16 2019
STATUS
approved