A238971 The number of nodes at odd level in divisor lattice in canonical order.

0, 1, 1, 2, 2, 3, 4, 2, 4, 4, 6, 8, 3, 5, 6, 8, 9, 12, 16, 3, 6, 7, 10, 8, 12, 16, 13, 18, 24, 32, 4, 7, 9, 12, 10, 15, 20, 16, 18, 24, 32, 27, 36, 48, 64, 4, 8, 10, 14, 12, 18, 24, 12, 20, 22, 30, 40, 24, 32, 36, 48, 64, 40, 54, 72, 96, 128

Offset

0,4

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) = A056924(A063008(n,k)).

T(n,k) = A238963(n,k) - A238970(n,k).

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

Example

Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  2, 3, 4;
  2, 4, 4,  6, 8;
  3, 5, 6,  8, 9, 12, 16;
  3, 6, 7, 10, 8, 12, 16, 13, 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-> floor(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)
b(n)={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. A238958 in canonical order.

Cf. A056924, A063008, A238963, A238970.

Sequence in context: A114892 A285705 A238958 * A194331 A372606 A290600

Adjacent sequences: A238968 A238969 A238970 * A238972 A238973 A238974

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