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A331023 Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).

%I A331023 #11 May 28 2021 00:56:07
%S A331023 1,1,1,2,1,1,1,3,2,1,1,4,1,1,1,5,1,4,1,4,1,1,1,7,2,1,3,4,1,1,1,7,1,1,
%T A331023 1,9,1,1,1,7,1,1,1,4,4,1,1,12,2,4,1,4,1,7,1,7,1,1,1,11,1,1,4,11,1,1,1,
%U A331023 4,1,1,1,16,1,1,4,4,1,1,1,12,5,1,1,11,1,1,1,7,1,11,1,4,1,1,1,19,1,4,4,9,1,1,1,7,1
%N A331023 Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).
%C A331023 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 A331023 Antti Karttunen, <a href="/A331023/b331023.txt">Table of n, a(n) for n = 1..65537</a>
%H A331023 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A331023 a(2^n) = A330994(n).
%t A331023 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A331023 Table[Length[facs[n]]/Length[Select[facs[n],UnsameQ@@#&]],{n,100}]//Numerator
%o A331023 (PARI)
%o A331023 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 A331023 A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));
%o A331023 A331023(n) = numerator(A001055(n)/A045778(n)); \\ _Antti Karttunen_, May 27 2021
%Y A331023 Positions of 1's are A005117.
%Y A331023 Positions of 2's appear to be A001248.
%Y A331023 The denominators are A331024.
%Y A331023 The rounded quotients are A331048.
%Y A331023 The same for integer partitions is A330994.
%Y A331023 Cf. A001055, A001222, A002033, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.
%K A331023 nonn,frac
%O A331023 1,4
%A A331023 _Gus Wiseman_, Jan 08 2020
%E A331023 More terms from _Antti Karttunen_, May 27 2021