OFFSET
1,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = (p+3)^e.
If n = Product p(k)^e(k) then a(n) = Product (p(k)+3)^e(k).
From Vaclav Kotesovec, Feb 11 2023: (Start)
Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(1-s) - 3*p^(-s)).
Dirichlet g.f.: zeta(s-1) * (1 + 3/(2^s - 5)) * Product_{p prime, p>2} (1 + 3/(p^s - p - 3)).
Sum_{k=1..n} a(k) has average order 3 * c * zeta(r-1) * n^r / (5*log(5)), where r = log(5)/log(2) = 2.321928094... and c = Product_{p prime, p>2} (1 + 3/(p^r - p - 3)) = 1.68551448153095... (End)
MATHEMATICA
f[p_, e_] := (p + 3)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Vaclav Kotesovec, Feb 11 2023 *)
PROG
(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i, 1] += 3); factorback(f); \\ Michel Marcus, Jun 09 2014
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-3*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2023
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Jaroslav Krizek, Oct 17 2009
EXTENSIONS
More terms from Michel Marcus, Jun 09 2014
STATUS
approved