A238964 Size of divisor lattice in canonical order.

0, 1, 2, 4, 3, 7, 12, 4, 10, 12, 20, 32, 5, 13, 17, 28, 33, 52, 80, 6, 16, 22, 36, 24, 46, 72, 54, 84, 128, 192, 7, 19, 27, 44, 31, 59, 92, 64, 75, 116, 176, 135, 204, 304, 448, 8, 22, 32, 52, 38, 72, 112, 40, 82, 96, 148, 224, 104, 160, 186, 280, 416, 216, 324, 480, 704, 1024

Offset: 0

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
  0;
  1;
  2,  4;
  3,  7, 12;
  4, 10, 12, 20, 32;
  5, 13, 17, 28, 33, 52, 80;
  6, 16, 22, 36, 24, 46, 72, 54, 84, 128, 192;
  ...
		

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. A238953 in canonical order.

Cf. A062799, A063008.

Programs

Maple

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-> (p-> add(nops(factorset(d)), d=divisors
    (p)))(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 24 2020

Mathematica

A062799[n_] := PrimeNu[Divisors[n]] // Total;
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
A063008[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T[n_] := A062799 /@ A063008[n];
Table[T[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Apr 16 2025, after Alois P. Heinz in A063008 *)

PARI

\\ here b(n) is A062799.
b(n)={sumdiv(n, d, omega(d))}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{concat(vector(9, n, Row(n-1)))} \\ Andrew Howroyd, Mar 24 2020

Formula

T(n,k) = A062799(A063008(n,k)). - Andrew Howroyd, Mar 24 2020

Extensions

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 24 2020