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A129628 Inverse Moebius transform of A001511.

%I A129628 #31 Jul 30 2025 00:52:50
%S A129628 1,3,2,6,2,6,2,10,3,6,2,12,2,6,4,15,2,9,2,12,4,6,2,20,3,6,4,12,2,12,2,
%T A129628 21,4,6,4,18,2,6,4,20,2,12,2,12,6,6,2,30,3,9,4,12,2,12,4,20,4,6,2,24,
%U A129628 2,6,6,28,4,12,2,12,4,12,2,30,2,6,6,12,4,12,2,30,5,6,2,24,4,6,4,20
%N A129628 Inverse Moebius transform of A001511.
%C A129628 Dirichlet convolution of A000005 with A209229. - _Ridouane Oudra_, Jul 25 2025
%H A129628 Amiram Eldar, <a href="/A129628/b129628.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)
%F A129628 a(2n) = a(n) + A000005(2n), a(2n+1) = A000005(2n+1).
%F A129628 Dirichlet g.f.: zeta(s)^2 * 2^s/(2^s-1). - _Ralf Stephan_, Jun 17 2007, corrected by _Vaclav Kotesovec_, Feb 02 2019
%F A129628 a(n) = Sum_{d|n} A001511(d). - _Andrew Howroyd_, Aug 04 2018
%F A129628 Sum_{k=1..n} a(k) ~ 2*n * (2*gamma - 1 + log(n/2)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Feb 02 2019
%F A129628 Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = e+1 for p > 2. - _Amiram Eldar_, Sep 30 2020
%F A129628 From _Ridouane Oudra_, Sep 30 2024: (Start)
%F A129628 a(n) = Sum_{i=0..A007814(n)} tau(n/2^i).
%F A129628 a(n) = Sum_{d|2*n} A007814(d).
%F A129628 a(n) = (1/2)*A001511(n)*A099777(n).
%F A129628 a(n) = (1/2)*(A001511(n) + 1)*A000005(n).
%F A129628 a(n) = A115364(n)*A001227(n). (End)
%p A129628 seq(add(padic[ordp](2*d, 2), d in numtheory[divisors](n)), n=1..100); # _Ridouane Oudra_, Sep 30 2024
%t A129628 f[p_, e_] := If[p==2, (e+1)*(e+2)/2, e+1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 30 2020 *)
%o A129628 (PARI) a(n)={sumdiv(n, d, 1 + valuation(d, 2))} \\ _Andrew Howroyd_, Aug 04 2018
%Y A129628 Row sums of A129353.
%Y A129628 Cf. A000005, A001511, A001620, A129628.
%Y A129628 Cf. A007814, A099777, A115364, A001227, A209229.
%K A129628 nonn,mult,easy
%O A129628 1,2
%A A129628 _Ralf Stephan_, May 31 2007