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A096825 Maximal size of an antichain in divisor lattice D(n). 15
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 (list; graph; refs; listen; history; text; internal format)
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

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

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-Fran├žois 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

Cf. A001055, A008480, A096826, A096827, A253249, A285573.

Sequence in context: A064372 A343943 A345926 * A318369 A007875 A323437

Adjacent sequences:  A096822 A096823 A096824 * A096826 A096827 A096828

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

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Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)