A238971 -id:A238971 - cp's OEIS frontend

A238958 The number of nodes at odd level in divisor lattice in graded colexicographic order.

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

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			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, 8, 10, 12, 13, 16, 18, 24, 32;
  ...

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.

Cf. A056924 in graded colexicographic order.

Cf. A036035, A074139, A238957, A238971.

\\ here b(n) is A056924.
b(n)={numdiv(n)\2}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 01 2020

T(n,k) = A056924(A036035(n,k)).
From Andrew Howroyd, Apr 01 2020: (Start)
T(n,k) = A074139(n,k) - A238957(n,k).
T(n,k) = floor(A074139(n,k)/2). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 01 2020

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

Original entry on oeis.org

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

Sung-Hyuk Cha, Mar 07 2014

			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;
  ...

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.

Cf. A238957 in canonical order.
Leftmost column gives A008619.
Last terms of rows give A011782.
Cf. A038548, A063008, A238963, A238971.

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

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 *)

\\ 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

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)

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

cp's OEIS Frontend

A238958 The number of nodes at odd level in divisor lattice in graded colexicographic order.

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