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

0,3

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

T(n,k) = A074206(A063008(n,k)). - Andrew Howroyd, Apr 26 2020

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

Row sums are A035341.

Cf. A238962 in canonical order, A002033.

Cf. A063008, A074206.

Sequence in context: A272615 A356188 A238962 * A098348 A131420 A095705

Adjacent sequences: A238972 A238973 A238974 * A238976 A238977 A238978

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