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

0,3

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.

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)

Example

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;
  ...

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

Prog

(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

Crossrefs

Cf. A238957 in canonical order.

Cf. A038548, A063008, A238963, A238971.

Sequence in context: A112184 A112213 A238957 * A085755 A330216 A241952

Adjacent sequences: A238967 A238968 A238969 * A238971 A238972 A238973

Keyword

nonn,tabf

Author

Sung-Hyuk Cha, Mar 07 2014

Extensions

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

Status

approved