cp's OEIS Frontend

A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

%I A336569 #14 Sep 02 2020 23:05:13
%S A336569 1,1,1,1,1,0,1,1,1,0,1,2,1,0,0,1,1,2,1,2,0,0,1,3,1,0,1,2,1,0,1,1,0,0,
%T A336569 0,0,1,0,0,3,1,0,1,2,2,0,1,4,1,2,0,2,1,3,0,3,0,0,1,0,1,0,2,1,0,0,1,2,
%U A336569 0,0,1,5,1,0,2,2,0,0,1,4,1,0,1,0,0,0,0
%N A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).
%C A336569 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
%e A336569 The a(n) chains for n = 12, 72, 144, 192 (ones not shown):
%e A336569   12/3    72/18/2       144/72/18/2       192/96/48/24/12/3
%e A336569   12/4/2  72/18/9/3     144/72/18/9/3     192/64/32/16/8/4/2
%e A336569           72/24/12/3    144/48/24/12/3    192/96/32/16/8/4/2
%e A336569           72/24/8/4/2   144/72/24/12/3    192/96/48/16/8/4/2
%e A336569           72/24/12/4/2  144/48/16/8/4/2   192/96/48/24/8/4/2
%e A336569                         144/48/24/8/4/2   192/96/48/24/12/4/2
%e A336569                         144/72/24/8/4/2
%e A336569                         144/48/24/12/4/2
%e A336569                         144/72/24/12/4/2
%t A336569 strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
%t A336569 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
%t A336569 strchs[n_]:=If[n==1,{{}},If[!strsigQ[n],{},Join@@Table[Prepend[#,d]&/@strchs[d],{d,Select[Most[Divisors[n]],strsigQ]}]]];
%t A336569 Table[Length[fasmax[strchs[n]]],{n,100}]
%Y A336569 A336423 is the non-maximal version.
%Y A336569 A336570 is the version for chains not necessarily containing n.
%Y A336569 A000005 counts divisors.
%Y A336569 A001055 counts factorizations.
%Y A336569 A001222 counts prime factors with multiplicity.
%Y A336569 A007425 counts divisors of divisors.
%Y A336569 A032741 counts proper divisors.
%Y A336569 A045778 counts strict factorizations.
%Y A336569 A071625 counts distinct prime multiplicities.
%Y A336569 A074206 counts strict chains of divisors from n to 1.
%Y A336569 A130091 lists numbers with distinct prime multiplicities.
%Y A336569 A181796 counts divisors with distinct prime multiplicities.
%Y A336569 A253249 counts chains of divisors.
%Y A336569 A336422 counts divisible pairs of divisors, both in A130091.
%Y A336569 A336424 counts factorizations using A130091.
%Y A336569 A336571 counts divisor sets of elements of A130091.
%Y A336569 Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336425, A336500, A336568.
%K A336569 nonn
%O A336569 1,12
%A A336569 _Gus Wiseman_, Jul 29 2020