A238963 Number of divisors of A063008(n,k).

1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64, 8, 14, 18, 24, 20, 30, 40, 32, 36, 48, 64, 54, 72, 96, 128, 9, 16, 21, 28, 24, 36, 48, 25, 40, 45, 60, 80, 48, 64, 72, 96, 128, 81, 108, 144, 192, 256, 10, 18, 24, 32, 28, 42, 56, 30, 48, 54, 72, 96, 50, 60, 80, 90, 120, 160, 64, 96, 128, 108, 144, 192, 256, 162, 216, 288, 384, 512

Offset

0,2

Comments

Equivalent to A074139 but using canonical order.

Links

Alois P. Heinz, Rows n = 0..30, flattened

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

Formula

T(n, k) = A000005(A063008(n,k)).

Trow(n) = List_{p in Partitions(n)} (Product_{t in p}(t + 1)). # Peter Luschny, Dec 11 2023

Example

Triangle begins:
  1;
  2;
  3,  4;
  4,  6,  8;
  5,  8,  9, 12, 16;
  6, 10, 12, 16, 18, 24, 32;
  7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64;
  ...

Maple

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-> numtheory[tau](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

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]]];
T[n_] := DivisorSigma[0, #]&[Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n]];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

Prog

(PARI) \\ here b(n) is A000005.
b(n)={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 24 2020
(SageMath)
def A238963row(n):
    return list(product(t + 1 for t in p) for p in Partitions(n))
print([A238963row(n) for n in range(10)])  # Peter Luschny, Dec 11 2023

Crossrefs

Row sums are A074141.

Cf. A000005, A000041, A063008, A074139.

Sequence in context: A241088 A074139 A355026 * A342940 A331527 A326575

Adjacent sequences: A238960 A238961 A238962 * A238964 A238965 A238966

Keyword

nonn,look,tabf

Author

Sung-Hyuk Cha, Mar 07 2014

Extensions

Offset corrected by Andrew Howroyd, Mar 24 2020

Status

approved