A238963 - OEIS

%I #26 Dec 13 2023 15:53:16

%S 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,

%T 36,48,64,8,14,18,24,20,30,40,32,36,48,64,54,72,96,128,9,16,21,28,24,

%U 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

%N Number of divisors of A063008(n,k).

%C Equivalent to A074139 but using canonical order.

%H Alois P. Heinz, <a href="/A238963/b238963.txt">Rows n = 0..30, flattened</a>

%H S.-H. Cha, E. G. DuCasse, and L. V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arxiv:1405.5283

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

%F Trow(n) = List_{p in Partitions(n)} (Product_{t in p}(t + 1)). # _Peter Luschny_, Dec 11 2023

%e Triangle begins:

%e   1;

%e   2;

%e   3,  4;

%e   4,  6,  8;

%e   5,  8,  9, 12, 16;

%e   6, 10, 12, 16, 18, 24, 32;

%e   7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64;

%e   ...

%p b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->

%p     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):

%p T:= n-> map(x-> numtheory[tau](mul(ithprime(i)

%p         ^x[i], i=1..nops(x))), b(n$2))[]:

%p seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 24 2020

%o (PARI) \\ here b(n) is A000005.

%o b(n)={numdiv(n)}

%o N(sig)={prod(k=1, #sig, prime(k)^sig[k])}

%o Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}

%o { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 24 2020

%o (SageMath)

%o def A238963row(n):

%o     return list(product(t + 1 for t in p) for p in Partitions(n))

%o print([A238963row(n) for n in range(10)])  # _Peter Luschny_, Dec 11 2023

%Y Row sums are A074141.

%Y Cf. A000005, A000041, A063008, A074139.

%K nonn,tabf

%O 0,2

%A _Sung-Hyuk Cha_, Mar 07 2014

%E Offset corrected by _Andrew Howroyd_, Mar 24 2020