A238970 The number of nodes at even level in divisor lattice in canonical order.

1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 3, 5, 6, 8, 9, 12, 16, 4, 6, 8, 10, 8, 12, 16, 14, 18, 24, 32, 4, 7, 9, 12, 10, 15, 20, 16, 18, 24, 32, 27, 36, 48, 64, 5, 8, 11, 14, 12, 18, 24, 13, 20, 23, 30, 40, 24, 32, 36, 48, 64, 41, 54, 72, 96, 128

Offset: 0

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
  1;
  1;
  2, 2;
  2, 3, 4;
  3, 4, 5,  6, 8;
  3, 5, 6,  8, 9, 12, 16;
  4, 6, 8, 10, 8, 12, 16, 14, 18, 24, 32;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
• 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.

Crossrefs

Cf. A238957 in canonical order.

Leftmost column gives A008619.

Last terms of rows give A011782.

Cf. A038548, A063008, A238963, A238971.

Programs

Maple

b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> ceil(numtheory[tau](mul(ithprime(i)
        ^x[i], i=1..nops(x)))/2), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 25 2020

Mathematica

A063008row[n_] := Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n];
A038548[n_] := Ceiling[DivisorSigma[0, n]/2];
T[n_] := A038548 /@ A063008row[n];
Table[T[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Jan 30 2025 *)

PARI

\\ here b(n) is A038548.
b(n)={ceil(numdiv(n)/2)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

Formula

From Andrew Howroyd, Mar 25 2020: (Start)

T(n,k) = A038548(A063008(n,k)).

T(n,k) = A238963(n,k) - A238971(n,k).

T(n,k) = ceiling(A238963(n,k)/2). (End)

Extensions

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 25 2020