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

0,3

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

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

Example

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;
  ...

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

Prog

(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

Crossrefs

Cf. A238961 in canonical order.

Cf. A063008, A238952.

Sequence in context: A086187 A088082 A238961 * A139013 A108337 A285534

Adjacent sequences: A238971 A238972 A238973 * A238975 A238976 A238977

Keyword

nonn,tabf

Author

Sung-Hyuk Cha, Mar 07 2014

Extensions

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

Status

approved