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A096825
Maximal size of an antichain in divisor lattice D(n).
18
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3
OFFSET
1,6
COMMENTS
The divisor lattice D(n) is the lattice of the divisors of the natural number n.
Also the number of divisors of n with half (rounded either way) as many prime factors (counting multiplicity) as n. - Gus Wiseman, Aug 24 2018
LINKS
S.-H. Cha, E. G. DuCasse and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT] (2014), (2.19).
FORMULA
a(n) is the coefficient at x^k in (1+x+...+x^k_1)*...*(1+x+...+x^k_q) where n=p_1^k_1*...*p_q^k_q is the prime factorization of n and k=floor((k_1+...+k_q)/2). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 22 2004
EXAMPLE
There are two maximal size antichains of divisors of 180, namely {12, 18, 20, 30, 45} and {4, 6, 9, 10, 15}. Both have length 5 so a(180) = 5. - Gus Wiseman, Aug 24 2018
MAPLE
a:=proc(n) local klist, x; klist:=ifactors(n)[2, 1..-1, 2]; coeff(normal(mul((1-x^(k+1))/(1-x), k=klist)), x, floor(add(k, k=klist)/2)) end: seq(a(n), n=1..100);
MATHEMATICA
a[n_] := Module[{pp, kk, x}, {pp, kk} = Transpose[FactorInteger[n]]; Coefficient[ Product[ Total[x^Range[0, k]], {k, kk}], x, Quotient[ Total[ kk], 2] ] ]; Array[a, 100] (* Jean-François Alcover, Nov 20 2017 *)
Table[Length[Select[Divisors[n], PrimeOmega[#]==Round[PrimeOmega[n]/2]&]], {n, 50}] (* Gus Wiseman, Aug 24 2018 *)
PROG
(Sage)
def A096825(n) :
if n==1 : return 1
R.<t> = QQ[]; mults = [x[1] for x in factor(n)]
return prod((t^(m+1)-1)//(t-1) for m in mults)[sum(mults)//2]
# Eric M. Schmidt, May 11 2013
(PARI) a(n)=if(n<6||isprimepower(n), return(1)); my(d=divisors(n), r=1, u); d=d[2..#d-1]; for(k=0, 2^#d-1, if(hammingweight(k)<=r, next); u=vecextract(d, k); for(i=1, #u, for(j=i+1, #u, if(u[j]%u[i]==0, next(3)))); r=#u); r \\ Charles R Greathouse IV, May 14 2013
(Python)
from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A096825(n):
fs = factorint(n)
return len(list(multiset_combinations(fs, sum(fs.values())//2))) # Chai Wah Wu, Aug 23 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com) and Vladeta Jovovic, Aug 17 2004
EXTENSIONS
More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 22 2004
STATUS
approved