A331023 - OEIS

%I #11 May 28 2021 00:56:07

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

%C 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 Antti Karttunen, <a href="/A331023/b331023.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(2^n) = A330994(n).

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[facs[n]]/Length[Select[facs[n],UnsameQ@@#&]],{n,100}]//Numerator

%o (PARI)

%o 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 A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));

%o A331023(n) = numerator(A001055(n)/A045778(n)); \\ _Antti Karttunen_, May 27 2021

%Y Positions of 1's are A005117.

%Y Positions of 2's appear to be A001248.

%Y The denominators are A331024.

%Y The rounded quotients are A331048.

%Y The same for integer partitions is A330994.

%Y Cf. A001055, A001222, A002033, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

%K nonn,frac

%O 1,4

%A _Gus Wiseman_, Jan 08 2020

%E More terms from _Antti Karttunen_, May 27 2021