1,2

Alois P. Heinz, Table of n, a(n) for n = 1..10000

a(n) = Sum_{d | n} A000688(d).

Multiplicative with a(p^e) = A000070(e). - Amiram Eldar, Sep 09 2020

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

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.

with(combinat): with(numtheory):

a:= n-> add(mul(numbpart(i[2]), i=ifactors(d)[2]), d=divisors(n)):

seq(a(n), n=1..100); # Alois P. Heinz, Apr 08 2011

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 *)

(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]))}

A188581(n)=sumdiv(n, d, A000688(d))

r=vector(66, n, A188581(n)) /* show terms */ /* Joerg Arndt, Apr 08 2011 */

Cf. A000688, A000070.

Sequence in context: A101113 A055155 A085191 * A347958 A318316 A328721

Adjacent sequences: A188578 A188579 A188580 * A188582 A188583 A188584

nonn,easy,mult

Marc Bogaerts, Apr 04 2011

approved