A238974 The size (the number of arcs) in the transitive closure of divisor lattice in canonical order.

0, 1, 3, 5, 6, 12, 19, 10, 22, 27, 42, 65, 15, 35, 48, 74, 90, 138, 211, 21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665, 28, 70, 108, 165, 130, 240, 365, 268, 324, 492, 746, 594, 900, 1362, 2059, 36, 92, 147, 224, 186, 342, 519, 200, 410, 495, 750, 1135, 552, 836, 1008, 1524, 2302, 1215, 1836, 2772, 4182, 6305

Offset: 0

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
   0;
   1;
   3,  5;
   6, 12, 19;
  10, 22, 27,  42, 65;
  15, 35, 48,  74, 90, 138, 211;
  21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665;
  ...
		

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. A238961 in canonical order.

Cf. A063008, A238952.

Programs

Maple

with(numtheory):
f:= n-> add(tau(d), d=divisors(n) minus {n}):
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-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 26 2020

PARI

\\ here b(n) is A238952.
b(n) = {sumdivmult(n, d, numdiv(d)) - numdiv(n)}
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 26 2020

Formula

T(n,k) = A238952(A063008(n,k)). - Andrew Howroyd, Mar 26 2020

Extensions

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 26 2020