OFFSET
0,2
COMMENTS
A partition of n is perfect if it contains just one partition of every number less than n when repeated parts are regarded as indistinguishable.
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56. It is in the sequence because the partition [1,1,1,4] is perfect.
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p 123.
LINKS
Alois P. Heinz, Rows n = 0..500, flattened
EXAMPLE
54 = 2*3*3*3 is in the sequence because the partition [1,2,2,2] is perfect.
24 = 2*2*2*3 is not in the sequence because the partition [1,1,1,2] is not perfect (1+1+1=1+2; it is complete).
Triangle T(n,k) begins:
1;
2;
4;
6, 8;
16;
18, 20, 32;
64;
42, 54, 56, 128;
...
MAPLE
with(numtheory):
T:= proc(m) local b, ll, p;
if m=0 then return 1 fi;
p:= proc(l) ll:=ll, 2^(l[1]-1)*mul(ithprime(
mul(l[j], j=1..i-1))^(l[i]-1), i=2..nops(l)) end:
b:= proc(n, l) `if`(n=1, p(l), seq(b(n/d, [l[], d])
, d=divisors(n) minus{1})) end:
ll:= NULL; b(m+1, []): sort([ll])[]
end:
seq(T(n), n=0..20); # Alois P. Heinz, Jun 08 2015
MATHEMATICA
T[0] = {1}; T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, 2^(l[[1]]-1)*Product[Prime[Product[l[[j]], {j, 1, i-1}]]^(l[[i]]-1), {i, 2, Length[l]}]]; 1); b[n_, l_List] := If[n == 1, p[l], Table[b[n/d, Append[l, d]], {d, Divisors[n] // Rest}]]; ll = {}; b[m+1, {}]; Sort[ll] ]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 28 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Emeric Deutsch, Jun 07 2015
STATUS
approved