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A238963
Number of divisors of A063008(n,k).
9
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
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
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.
Sequence in context: A241088 A074139 A355026 * A342940 A331527 A326575
KEYWORD
nonn,tabf
AUTHOR
Sung-Hyuk Cha, Mar 07 2014
EXTENSIONS
Offset corrected by Andrew Howroyd, Mar 24 2020
STATUS
approved