OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
Sung-Hyuk Cha, Edgar G. DuCasse, Louis V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], (2014).
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See H(n) page 3.
FORMULA
EXAMPLE
Triangle T(n,k) begins:
1;
1;
2, 3;
4, 8, 13;
8, 20, 26, 44, 75;
16, 48, 76, 132, 176, 308, 541;
32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683;
...
MAPLE
g:= proc(n) option remember; (1+add(g(n/d),
d=numtheory[divisors](n) minus {1, n}))
end:
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-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9); # Alois P. Heinz, Apr 26 2020
MATHEMATICA
(* b is A074206 *)
b[n_] := b[n] = If[n < 2, n, b /@ Most[Divisors[n]] // Total];
T[n_] := b /@ (Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]);
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jan 04 2021 *)
PROG
(PARI) \\ here b(n) is A074206.
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
b(n)={if(!n, 0, my(sig=factor(n)[, 2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r, k)*(-1)^(r-k))))}
Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Aug 30 2020
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Sung-Hyuk Cha, Mar 07 2014
EXTENSIONS
Offset changed and terms a(42) and beyond from Andrew Howroyd, Apr 26 2020
STATUS
approved