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A188581
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Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.
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2
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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, 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, 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
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OFFSET
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1,2
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s)^2 * Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
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
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EXAMPLE
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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.
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MAPLE
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with(combinat): with(numtheory):
a:= n-> add(mul(numbpart(i[2]), i=ifactors(d)[2]), d=divisors(n)):
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MATHEMATICA
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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 *)
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 *)
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PROG
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(GAP)trf:=function ( f, x ) # the Dirichlet convolution 1 * f
local d;
d := DivisorsInt( x );
return Sum( d, function ( i )
return f( i );
end );
end;
nra:=function ( x ) # the number of Abelian Groups of order(n)
local pp, ll;
pp := PrimePowersInt( x );
ll := [ 1 .. Size( pp ) / 2 ];
return Product( List( 2 * ll, function ( i )
return NrPartitions( pp[i] );
end ) );
end;
a:=function ( n )
return trf( nra, n );
end;
(PARI)
A000688(n)={local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))}
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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