OFFSET
0,2
COMMENTS
A partition of n is complete if every number from 1 to n can be represented as a sum of parts of the partition.
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 complete.
Except for a(0)=1, there are no odd numbers in the sequence. Indeed, a partition having an odd Heinz number does not have 1 as a part and, consequently, it cannot be complete.
LINKS
Alois P. Heinz, Rows n = 0..30, flattened
SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.
EXAMPLE
54 = 2*3*3*3 is in the sequence because the partition [1,2,2,2] is complete.
28 = 2*2*7 is not in the sequence because the partition [1,1,4] is not complete.
Triangle T(n,k) begins:
1;
2;
4;
6, 8;
12, 16;
18, 20, 24, 32;
30, 36, 40, 48, 64;
42, 54, 56, 60, 72, 80, 96, 128;
84, 90, 100, 108, 112, 120, 144, 160, 192, 256;
...
MAPLE
T:= proc(m) local b, ll, p;
p:= proc(l) ll:=ll, (mul(ithprime(j), j=l)); 1 end:
b:= proc(n, i, l) `if`(i<2, p([l[], 1$n]), `if`(n<2*i-1,
b(n, iquo(n+1, 2), l), b(n, i-1, l)+b(n-i, i, [l[], i])))
end: ll:= NULL; b(m, iquo(m+1, 2), []): sort([ll])[]
end:
seq(T(n), n=0..12); # Alois P. Heinz, Jun 07 2015
MATHEMATICA
T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, Product[Prime[j], {j, l}]]; 1); b[n_, i_, l_List] := If[i<2, p[Join[l, Array[1&, n]]], If[n < 2*i-1, b[n, Quotient[n+1, 2], l], b[n, i-1, l] + b[n-i, i, Append[l, i] ]]]; ll = {}; b[m, Quotient[m+1, 2], {}]; Sort[ll]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Emeric Deutsch, Jun 07 2015
STATUS
approved