A238975 Number of perfect partitions in canonical order.
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, 64, 256, 544, 976, 768, 1888, 3408, 2316, 3172, 5740, 10404, 7880, 14300, 25988, 47293, 128, 576, 1376, 2496, 2208, 5536, 10096, 2568
Offset: 0
Examples
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; ...
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.
Programs
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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 *)
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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
Formula
Extensions
Offset changed and terms a(42) and beyond from Andrew Howroyd, Apr 26 2020