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A096825 Maximal size of an antichain in divisor lattice D(n). 5
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, 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]

end # 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

CROSSREFS

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

Sequence in context: A079553 A001221 A064372 * A318369 A007875 A259936

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 December 14 04:53 EST 2018. Contains 318090 sequences. (Running on oeis4.)