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

%I A166632 #30 Feb 05 2025 05:51:02
%S A166632 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,
%T A166632 48,64,48,56,64,60,32,80,64,96,64,72,72,96,64,80,96,84,80,128,88,92,
%U A166632 64,144,128
%N A166632 Totally multiplicative sequence with a(p) = 2*(p-1) for prime p.
%H A166632 G. C. Greubel, <a href="/A166632/b166632.txt">Table of n, a(n) for n = 1..10000</a>
%H A166632 Vaclav Kotesovec, <a href="/A166632/a166632.jpg">Graph - the asymptotic ratio</a>.
%F A166632 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).
%F A166632 a(n) = A061142(n) * A003958(n) = 2^bigomega(n) * A003958(n) = 2^A001222(n) * A003958(n).
%F A166632 Dirichlet g.f.: Product_{p prime} 1/(1 - 2*(p-1)*p^(-s)). - _Robert Israel_, May 19 2016
%F A166632 From _Vaclav Kotesovec_, Mar 08 2023: (Start)
%F A166632 Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - (2 - p^(2-s))/(p^s-2*p+2)).
%F A166632 Let f(s) = Product_{p prime} (1 - (2 - p^(2-s)) / (p^s - 2*p + 2)).
%F A166632 Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
%F A166632 f(2) = Product_{p prime} (1 - 2/(p^2 - 2*p + 2)) = 0.353804459718477500968617797456682002952375753701841967763205003892191...,
%F A166632 f'(2) = f(2) * Sum_{p prime} 2*log(p) / ((p-1) * (p^2 - 2*p + 2)) = 0.350193097012820163529213089258238034020398107720137317340667886409682...
%F A166632  and gamma is the Euler-Mascheroni constant A001620. (End)
%p A166632 f:= proc(n) local f;
%p A166632   mul((2*(f[1]-1))^f[2], f = ifactors(n)[2])
%p A166632 end proc:
%p A166632 map(f, [$1..100]); # _Robert Israel_, May 19 2016
%t A166632 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 *)
%t A166632 f[p_, e_] := (2*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 17 2023 *)
%o A166632 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*(p-1)*X))[n], ", ")) \\ _Vaclav Kotesovec_, Mar 08 2023
%Y A166632 Cf. A001222, A001620, A003958, A061142, A069205.
%K A166632 nonn,easy,mult,look
%O A166632 1,2
%A A166632 _Jaroslav Krizek_, Oct 18 2009