login
A336569
Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).
13
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, 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, 0, 0, 1, 5, 1, 0, 2, 2, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 0
OFFSET
1,12
COMMENTS
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.
EXAMPLE
The a(n) chains for n = 12, 72, 144, 192 (ones not shown):
12/3 72/18/2 144/72/18/2 192/96/48/24/12/3
12/4/2 72/18/9/3 144/72/18/9/3 192/64/32/16/8/4/2
72/24/12/3 144/48/24/12/3 192/96/32/16/8/4/2
72/24/8/4/2 144/72/24/12/3 192/96/48/16/8/4/2
72/24/12/4/2 144/48/16/8/4/2 192/96/48/24/8/4/2
144/48/24/8/4/2 192/96/48/24/12/4/2
144/72/24/8/4/2
144/48/24/12/4/2
144/72/24/12/4/2
MATHEMATICA
strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
strchs[n_]:=If[n==1, {{}}, If[!strsigQ[n], {}, Join@@Table[Prepend[#, d]&/@strchs[d], {d, Select[Most[Divisors[n]], strsigQ]}]]];
Table[Length[fasmax[strchs[n]]], {n, 100}]
CROSSREFS
A336423 is the non-maximal version.
A336570 is the version for chains not necessarily containing n.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336571 counts divisor sets of elements of A130091.
Sequence in context: A340653 A334744 A136567 * A324904 A109708 A035468
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2020
STATUS
approved