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%I #12 May 26 2017 04:26:45
%S 1,1,1,1,1,3,1,3,1,3,1,6,1,3,3,4,1,6,1,6,3,3,1,13,1,3,3,6,1,12,1,7,3,
%T 3,3,15,1,3,3,13,1,12,1,6,6,3,1,25,1,6,3,6,1,13,3,13,3,3,1,31,1,3,6,
%U 12,3,12,1,6,3,12,1,37,1,3,6,6,3,12,1,25,4,3,1,31,3,3,3,13,1,31,3,6,3,3
%N Number of ways to factor n into distinct factors with one level of parentheses.
%C Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
%C a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
%H R. J. Mathar, <a href="/A050345/b050345.txt">Table of n, a(n) for n = 1..2519</a>
%F Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
%F a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - _R. J. Mathar_, May 25 2017
%F a(n) = A050346(A101296(n)). - _Antti Karttunen_, May 25 2017
%e 12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
%Y Cf. A045778, A050346-A050350. a(p^k)=A050342. a(A002110)=A000258.
%K nonn
%O 1,6
%A _Christian G. Bower_, Oct 15 1999
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