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

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio.

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A333194 A349131 A366665 * A116596 A248692 A048656

Adjacent sequences: A166629 A166630 A166631 * A166633 A166634 A166635

Keyword

nonn,easy,mult,look

Author

Jaroslav Krizek, Oct 18 2009

Status

approved