OFFSET
1,16
COMMENTS
Dirichlet convolution product of A000688 with the Moebius function.
It appears that a(n) is nonzero for n in A001694, the powerful numbers. - T. D. Noe, Apr 06 2011 [This is correct: a(n) > 0 if and only if n is in A001694. - Amiram Eldar, Jun 10 2025]
There is a similar sequence defined by b(n) = Product_{i} floor(e(i)/2) where n = Product_{p} p(i)^e(i) is the usual prime factorization, which differs from a(n) at n = 64, 128, 256, 512, 576, 729,.... - R. J. Mathar, Sep 18 2012 [This sequence is A365550. - Amiram Eldar, Jun 10 2025]
The number of unordered factorizations of n into 1 and prime powers p^e where p is prime and e >= 2 (A025475). - Amiram Eldar, Jun 10 2025
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
Dirichlet g.f.: Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = Product_{k>=3} zeta(k/2) = 10.0301441966843566206076085895839492473559217336... - Vaclav Kotesovec, Apr 22 2025
Multiplicative with a(p^e) = A002865(e). - Amiram Eldar, Jun 10 2025
MAPLE
with(numtheory): with(combinat):
a:= n-> add(mobius(n/d) *mul(numbpart(i[2]),
i=ifactors(d)[2]), d=divisors(n)):
seq(a(n), n=1..110); # Alois P. Heinz, Apr 07 2011
MATHEMATICA
MobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}]; Do[b[[i]] = Plus @@ (MoebiusMu[i/Divisors[i]] a[[Divisors[i]]]), {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; MobiusTransform[Array[A688, 100]] (* T. D. Noe, Apr 06 2011 *)
f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 10 2025 *)
PROG
(GAP)
mtrf:=function ( f, x ) # the Moebius inversion formula
local d;
d := DivisorsInt( x );
return Sum( d, function ( i )
return f( i ) * MoebiusMu( (x / i) );
end );
end;
nra:=function ( x ) # the number of Abelian groups of order x
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 mtrf( nra, n );
end;
(PARI) a(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1), factor(n)[, 2])); \\ Amiram Eldar, Jun 10 2025
(Python)
from math import prod
from sympy import partition, factorint
def A188585(n): return prod(partition(e)-partition(e-1) for e in factorint(n).values()) # Chai Wah Wu, Jun 10 2025
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Marc Bogaerts, Apr 04 2011
STATUS
approved
