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A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.

%I A188581 #34 Aug 21 2021 05:31:40
%S A188581 1,2,2,4,2,4,2,7,4,4,2,8,2,4,4,12,2,8,2,8,4,4,2,14,4,4,7,8,2,8,2,19,4,
%T A188581 4,4,16,2,4,4,14,2,8,2,8,8,4,2,24,4,8,4,8,2,14,4,14,4,4,2,16,2,4,8,30,
%U A188581 4,8,2,8,4,8,2,28,2,4,8,8,4,8,2,24,12,4,2,16,4,4,4,14,2,16
%N A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.
%H A188581 Alois P. Heinz, <a href="/A188581/b188581.txt">Table of n, a(n) for n = 1..10000</a>
%F A188581 a(n) = Sum_{d | n} A000688(d).
%F A188581 Multiplicative with a(p^e) = A000070(e). - _Amiram Eldar_, Sep 09 2020
%F A188581 Dirichlet g.f.: zeta(s)^2 * Product_{k>=2} zeta(k*s). - _Ilya Gutkovskiy_, Nov 03 2020
%F A188581 Sum_{k=1..n} a(k) ~ n*((log(n) + 2*gamma - 1)*f(1) + f'(1)), where f(1) = Product_{k>=2} zeta(k) = A021002 = 2.1955691982567064617939..., f'(1) = f(1) * Sum_{k>=2} k*zeta'(k)/zeta(k) = -5.0385164470942955610707128990779476296197... and gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Aug 21 2021
%e A188581 For n=8; the divisors of 8 are 1,2,4,8. There are 1,1,2,3 abelian groups of these orders respectively, so a(n) = 1+1+2+3 = 7.
%p A188581 with(combinat): with(numtheory):
%p A188581 a:= n-> add(mul(numbpart(i[2]), i=ifactors(d)[2]), d=divisors(n)):
%p A188581 seq(a(n), n=1..100);  # _Alois P. Heinz_, Apr 08 2011
%t A188581 InverseMobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}]; Do[b[[i]] = Plus @@ a[[Divisors[i]]], {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; InverseMobiusTransform[Array[A688, 100]] (* _T. D. Noe_, Apr 07 2011 *)
%t A188581 f[0] = 1; f[e_] := f[e] = f[e - 1] + PartitionsP[e]; a[1] = 1; a[n_] := Times @@ (f[Last[#]] & /@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 09 2020 *)
%o A188581 (GAP)trf:=function ( f, x )  # the Dirichlet convolution 1 * f
%o A188581     local  d;
%o A188581     d := DivisorsInt( x );
%o A188581     return Sum( d, function ( i )
%o A188581             return f( i );
%o A188581         end );
%o A188581 end;
%o A188581 nra:=function ( x )     # the number of Abelian Groups of order(n)
%o A188581     local  pp, ll;
%o A188581     pp := PrimePowersInt( x );
%o A188581     ll := [ 1 .. Size( pp ) / 2 ];
%o A188581     return Product( List( 2 * ll, function ( i )
%o A188581               return NrPartitions( pp[i] );
%o A188581           end ) );
%o A188581 end;
%o A188581 a:=function ( n )
%o A188581     return trf( nra, n );
%o A188581 end;
%o A188581 (PARI)
%o A188581 A000688(n)={local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))}
%o A188581 A188581(n)=sumdiv(n,d,A000688(d))
%o A188581 r=vector(66,n,A188581(n)) /* show terms */ /* _Joerg Arndt_, Apr 08 2011 */
%Y A188581 Cf. A000688, A000070.
%K A188581 nonn,easy,mult
%O A188581 1,2
%A A188581 _Marc Bogaerts_, Apr 04 2011