A065886 -id:A065886 - cp's OEIS frontend

A065887 Smallest number whose square is divisible by n!.

1, 1, 2, 6, 12, 60, 60, 420, 1680, 5040, 5040, 55440, 332640, 4324320, 8648640, 43243200, 172972800, 2940537600, 8821612800, 167610643200, 335221286400, 7039647014400, 14079294028800, 323823762662400, 647647525324800, 3238237626624000, 6476475253248000

Offset: 0

Henry Bottomley, Nov 27 2001

nonn

			a(10) = 5040 since 10! = 3628800 and the smallest square divisible by this is 25401600 = 3628800*7 = 5040^2.

Alois P. Heinz, Table of n, a(n) for n = 0..735 (first 101 terms from Kevin P. Thompson)

Cf. A000142, A019554, A055772, A065886.

a:= n-> mul(i[1]^ceil(i[2]/2), i=ifactors(n!)[2]):
seq(a(n), n=0..26);  # Alois P. Heinz, Jan 24 2022

f[p_, e_] := p^Ceiling[e/2]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30, 0] (* Amiram Eldar, Feb 11 2024 *)

a(n) = A019554(A000142(n)) = sqrt(A065886(n)) = A000142(n)/A055772(n).

Missing a(0) inserted, formula corrected, and a(25)-a(26) added by Kevin P. Thompson, Jan 24 2022

A250029 Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 36, 144, 400, 900, 3600, 11025, 28224, 78400, 254016, 705600, 2286144, 6350400, 25401600, 85377600, 250905600, 768398400, 3073593600, 10600761600, 32464832400, 129859329600, 456536705625

Offset: 0

Andrei Cretu, Nov 11 2014

nonn

The number of binary strings, counted up to isomorphism, that can be constructed by taking an equal number (n) of 0's and 1's and partitioning both the 0's and the 1's into m runs using the same partition, can be written as:
dualseq[partition[n]]=m!^2/(Prod_j(m_j!))^2
where m_j is the multiplicity of runs of length j.
The numbers satisfy the relations Sum_j(m_j)=m, Sum_j(j*m_j)=n.
The strings obtained in this manner are a subset of those in A247651.
Both the finest and coarsest partitions of n minimize dualseq[partition[n]]. In this sense, dualseq[partition[n]] is another relative measure of the complexity of the partition and the associated binary strings.
a[n] is the number of strings, counted up to isomorphism, that can be generated based on the partition that maximizes dualseq[partition[n]].

			n=0 gives the empty string.
n=1 and the only possible partition generate 01 (and the isomorphic 10).
For n=2, both possible partitions generate, up to isomorphism, 1 string, 0011 (1100), and respectively 0101 (1010).
For n=3, the optimal partition is {1,2}, generating, up to isomorphism, 4 strings: 001011 (110100), 001101 (110010), 010011 (101100) and 011001 (100110).
For n=4, the optimal partition is {1,1,2}, generating, up to isomorphism, 9 strings: 00101011 (11010100), 00101101 (11010010), 00110101 (11001010), 01001011 (10110100), 01001101 (10110010), 01010011 (10101100), 01011001 (10100110), 01100101 (10011010) and 01101001 (10010110).

Cf. A247651, A046952, A092266, A136404, A176471, A065886.

dualseq[p_]:=Factorial[Length[p]]^2/Apply[Times,Map[Factorial[Count[p,#1]]&,Range[Max[Length[p]]]]]^2
a[n_]:=Max[Map[dualseq,IntegerPartitions[n]]]
Table[a[n],{n,0,25}] (* after A130670 *)

a[n]=Max[m!^2/(Prod_j(m_j!))^2] where Sum_j(m_j)=m, Sum_j(j*m_j)=n, over all partitions of n.

cp's OEIS Frontend

A065887 Smallest number whose square is divisible by n!.

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