A166632 Totally multiplicative sequence with a(p) = 2*(p-1) for prime p.

1, 2, 4, 4, 8, 8, 12, 8, 16, 16, 20, 16, 24, 24, 32, 16, 32, 32, 36, 32, 48, 40, 44, 32, 64, 48, 64, 48, 56, 64, 60, 32, 80, 64, 96, 64, 72, 72, 96, 64, 80, 96, 84, 80, 128, 88, 92, 64, 144, 128

Offset: 1

Author

Jaroslav Krizek, Oct 18 2009

Keywords

• nonn
• easy
• mult
• look

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio.

Crossrefs

Cf. A001222, A001620, A003958, A061142, A069205.

Programs

Maple

f:= proc(n) local f;
  mul((2*(f[1]-1))^f[2], f = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, May 19 2016

Mathematica

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*2^(PrimeOmega[m]), {m, 1, 100}](* G. C. Greubel, May 19 2016, based on A003958 *)
f[p_, e_] := (2*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*(p-1)*X))[n], ", ")) \\ Vaclav Kotesovec, Mar 08 2023

Formula

Multiplicative with a(p^e) = (2*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-1))^e(k).

a(n) = A061142(n) * A003958(n) = 2^bigomega(n) * A003958(n) = 2^A001222(n) * A003958(n).

Dirichlet g.f.: Product_{p prime} 1/(1 - 2*(p-1)*p^(-s)). - Robert Israel, May 19 2016

From Vaclav Kotesovec, Mar 08 2023: (Start)

Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - (2 - p^(2-s))/(p^s-2*p+2)).

Let f(s) = Product_{p prime} (1 - (2 - p^(2-s)) / (p^s - 2*p + 2)).

Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where

f(2) = Product_{p prime} (1 - 2/(p^2 - 2*p + 2)) = 0.353804459718477500968617797456682002952375753701841967763205003892191...,

f'(2) = f(2) * Sum_{p prime} 2*log(p) / ((p-1) * (p^2 - 2*p + 2)) = 0.350193097012820163529213089258238034020398107720137317340667886409682...

and gamma is the Euler-Mascheroni constant A001620. (End)