A166591 - OEIS

%I #21 Feb 12 2023 02:08:02

%S 1,5,6,25,8,30,10,125,36,40,14,150,16,50,48,625,20,180,22,200,60,70,

%T 26,750,64,80,216,250,32,240,34,3125,84,100,80,900,40,110,96,1000,44,

%U 300,46,350,288,130,50,3750,100,320,120,400,56,1080,112,1250,132,160

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

%H Vaclav Kotesovec, <a href="/A166591/b166591.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p+3)^e.

%F If n = Product p(k)^e(k) then a(n) = Product (p(k)+3)^e(k).

%F From _Vaclav Kotesovec_, Feb 11 2023: (Start)

%F Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(1-s) - 3*p^(-s)).

%F Dirichlet g.f.: zeta(s-1) * (1 + 3/(2^s - 5)) * Product_{p prime, p>2} (1 + 3/(p^s - p - 3)).

%F 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)

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

%o (PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] += 3); factorback(f); \\ _Michel Marcus_, Jun 09 2014

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-3*X))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 10 2023

%K nonn,mult

%O 1,2

%A _Jaroslav Krizek_, Oct 17 2009

%E More terms from _Michel Marcus_, Jun 09 2014