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A331024 Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).

%I A331024 #11 May 28 2021 00:56:14
%S A331024 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,3,1,3,1,1,1,5,1,1,2,3,1,1,1,3,1,1,
%T A331024 1,5,1,1,1,5,1,1,1,3,3,1,1,7,1,3,1,3,1,5,1,5,1,1,1,9,1,1,3,4,1,1,1,3,
%U A331024 1,1,1,9,1,1,3,3,1,1,1,7,2,1,1,9,1,1,1,5,1,9,1,3,1,1,1,10,1,3,3,5,1,1,1,5,1
%N A331024 Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).
%C A331024 A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.
%H A331024 Antti Karttunen, <a href="/A331024/b331024.txt">Table of n, a(n) for n = 1..65537</a>
%H A331024 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A331024 a(2^n) = A330995(n).
%t A331024 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A331024 Table[Length[facs[n]]/Length[Select[facs[n],UnsameQ@@#&]],{n,100}]//Denominator
%o A331024 (PARI)
%o A331024 A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
%o A331024 A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));
%o A331024 A331024(n) = denominator(A001055(n)/A045778(n)); \\ _Antti Karttunen_, May 27 2021
%Y A331024 Positions of 1's include all elements of A001248 as well as A005117. The first position of a 1 that is not in A167207 is 128.
%Y A331024 The numerators are A331023.
%Y A331024 The rounded quotients are A331048.
%Y A331024 The same for integer partitions is A330995.
%Y A331024 Cf. A001055, A005117, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.
%K A331024 nonn,frac
%O A331024 1,8
%A A331024 _Gus Wiseman_, Jan 08 2020
%E A331024 More terms from _Antti Karttunen_, May 27 2021