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

Author

Jaroslav Krizek, Oct 17 2009

Keywords

• nonn
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Programs

Mathematica

f[p_, e_] := (p + 3)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Vaclav Kotesovec, Feb 11 2023 *)

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

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)

Extensions

More terms from Michel Marcus, Jun 09 2014