A166591 Totally multiplicative sequence with a(p) = p+3 for prime p.

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

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

Sequence in context: A047186 A324485 A335827 * A342610 A160529 A039572

Adjacent sequences: A166588 A166589 A166590 * A166592 A166593 A166594

Keyword

nonn,mult

Author

Jaroslav Krizek, Oct 17 2009

Extensions

More terms from Michel Marcus, Jun 09 2014

Status

approved