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
Examples
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; ...
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
Programs
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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 *) -
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
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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
Formula
Trow(n) = List_{p in Partitions(n)} (Product_{t in p}(t + 1)). # Peter Luschny, Dec 11 2023
Extensions
Offset corrected by Andrew Howroyd, Mar 24 2020
Comments