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A166591
Totally multiplicative sequence with a(p) = p+3 for prime p.
18
1, 5, 6, 25, 8, 30, 10, 125, 36, 40, 14, 150, 16, 50, 48, 625, 20, 180, 22, 200, 60, 70, 26, 750, 64, 80, 216, 250, 32, 240, 34, 3125, 84, 100, 80, 900, 40, 110, 96, 1000, 44, 300, 46, 350, 288, 130, 50, 3750, 100, 320, 120, 400, 56, 1080, 112, 1250, 132, 160
OFFSET
1,2
LINKS
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
Sequence in context: A047186 A324485 A335827 * A342610 A160529 A039572
KEYWORD
nonn,mult
AUTHOR
Jaroslav Krizek, Oct 17 2009
EXTENSIONS
More terms from Michel Marcus, Jun 09 2014
STATUS
approved