A238950 -id:A238950 - cp's OEIS frontend

A238951 The number of arcs from odd to even level vertices in divisor lattice D(n).

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 2, 2, 2, 0, 3, 0, 3, 2, 2, 0, 5, 1, 2, 1, 3, 0, 6, 0, 2, 2, 2, 2, 6, 0, 2, 2, 5, 0, 6, 0, 3, 3, 2, 0, 6, 1, 3, 2, 3, 0, 5, 2, 5, 2, 2, 0, 10, 0, 2, 3, 3, 2, 6, 0, 3, 2, 6, 0, 8, 0, 2, 3, 3, 2, 6, 0, 6, 2, 2, 0, 10, 2, 2

Offset: 1

Sung-Hyuk Cha, Mar 07 2014

nonn

R. J. Mathar, Table of n, a(n) for n = 1..1000
Sung-Hyuk 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 (see 12th line in Table 1).

Cf. A056924, A062799, A238950, A358769.

read("transforms"):
omega := [seq(A001221(n), n=1..1000)] ;
ones := [seq(1,n=1..1000)] ;
a062799 := DIRICHLET(ones,omega) ;
for n from 1 do
    a238951 := floor(op(n,a062799)/2) ;
    printf("%d %d\n",n,a238951) ;
end do: # R. J. Mathar, May 28 2017

a(n) = A062799(n) - A238950(n) = floor(A062799(n)/2). [Cha eqs. (2.34), (2.37)]

a(n) = (A062799(n) - A358769(n))/2. - Ridouane Oudra, May 17 2025

A238959 The number of arcs from even to odd level vertices in divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 6, 2, 5, 6, 10, 16, 3, 7, 9, 14, 17, 26, 40, 3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96, 4, 10, 14, 16, 22, 30, 32, 38, 46, 58, 68, 88, 102, 152, 224, 4, 11, 16, 19, 20, 26, 36, 41, 48, 52, 56, 74, 80, 93, 108, 112, 140, 162, 208, 240, 352, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  2, 4,  6;
  2, 5,  6, 10, 16;
  3, 7,  9, 14, 17, 26, 40;
  3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96;
  ...

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. A238950 in graded colexicographic order.

Cf. A036035, A238953, A238960, A238972.

T(n,k) = A238950(A036035(n,k)).
From Andrew Howroyd, Apr 25 2020: (Start)
T(n,k) = ceiling(A238953(n,k)/2).
T(n,k) = A238953(n,k) - A238960(n,k). (End)

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

A238972 The number of arcs from even to odd level vertices in divisor lattice in canonical order.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 6, 2, 5, 6, 10, 16, 3, 7, 9, 14, 17, 26, 40, 3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96, 4, 10, 14, 22, 16, 30, 46, 32, 38, 58, 88, 68, 102, 152, 224, 4, 11, 16, 26, 19, 36, 56, 20, 41, 48, 74, 112, 52, 80, 93, 140, 208, 108, 162, 240, 352, 512

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  2, 4,  6;
  2, 5,  6, 10, 16;
  3, 7,  9, 14, 17, 26, 40;
  3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96;
  ...

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. A238959 in canonical order.

Cf. A063008, A238950, A238964, A238973.

with(numtheory):
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((p-> add(nops(factorset(d)), d=divisors
    (p)))(mul(ithprime(i)^x[i], i=1..nops(x)))/2), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 28 2020

From Andrew Howroyd, Mar 28 2020: (Start)
T(n,k) = A238950(A063008(n,k)).
T(n,k) = A238964(n,k) - A238973(n,k).
T(n,k) = ceiling(A238964(n,k)/2). (End)

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 28 2020

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A238951 The number of arcs from odd to even level vertices in divisor lattice D(n).

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A238959 The number of arcs from even to odd level vertices in divisor lattice in graded colexicographic order.

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A238972 The number of arcs from even to odd level vertices in divisor lattice in canonical order.

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