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

%I A238963 #31 Jan 09 2025 08:38:47
%S A238963 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 A238963 36,48,64,8,14,18,24,20,30,40,32,36,48,64,54,72,96,128,9,16,21,28,24,
%U A238963 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 A238963 Number of divisors of A063008(n,k).
%C A238963 Equivalent to A074139 but using canonical order.
%H A238963 Alois P. Heinz, <a href="/A238963/b238963.txt">Rows n = 0..30, flattened</a>
%H A238963 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 A238963 T(n, k) = A000005(A063008(n,k)).
%F A238963 Trow(n) = List_{p in Partitions(n)} (Product_{t in p}(t + 1)). # _Peter Luschny_, Dec 11 2023
%e A238963 Triangle begins:
%e A238963   1;
%e A238963   2;
%e A238963   3,  4;
%e A238963   4,  6,  8;
%e A238963   5,  8,  9, 12, 16;
%e A238963   6, 10, 12, 16, 18, 24, 32;
%e A238963   7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64;
%e A238963   ...
%p A238963 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A238963     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A238963 T:= n-> map(x-> numtheory[tau](mul(ithprime(i)
%p A238963         ^x[i], i=1..nops(x))), b(n$2))[]:
%p A238963 seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 24 2020
%t A238963 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 A238963 T[n_] := DivisorSigma[0, #]&[Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n]];
%t A238963 T /@ Range[0, 9] // Flatten (* _Jean-François Alcover_, Jan 09 2025, after _Alois P. Heinz_ *)
%o A238963 (PARI) \\ here b(n) is A000005.
%o A238963 b(n)={numdiv(n)}
%o A238963 N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
%o A238963 Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o A238963 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 24 2020
%o A238963 (SageMath)
%o A238963 def A238963row(n):
%o A238963     return list(product(t + 1 for t in p) for p in Partitions(n))
%o A238963 print([A238963row(n) for n in range(10)])  # _Peter Luschny_, Dec 11 2023
%Y A238963 Row sums are A074141.
%Y A238963 Cf. A000005, A000041, A063008, A074139.
%K A238963 nonn,look,tabf
%O A238963 0,2
%A A238963 _Sung-Hyuk Cha_, Mar 07 2014
%E A238963 Offset corrected by _Andrew Howroyd_, Mar 24 2020