OFFSET
0,2
COMMENTS
Partitions are ordered first by sum. Then all partitions of n are viewed as exponent tuples on n variables and their corresponding monomials are ordered reverse lexicographically. This gives a canonical ordering: [] [1] [2,0] [1,1] [3,0,0] [2,1,0] [1,1,1] [4,0,0,0] [3,1,0,0] [2,2,0,0] [2,1,1,0] [1,1,1,1]... Rearrangement of A025487, A036035 etc.
Or, least integer of each prime signature; resorted in accordance with the integer partitions described in A080577. - Alford Arnold, Feb 13 2008
LINKS
Alois P. Heinz, Rows n = 0..30, flattened
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
FORMULA
bigomega(T(n,k)) = n. - Andrew Howroyd, Mar 28 2020
EXAMPLE
Partition [2,1,1,1] for n=5 gives 2^2*3*5*7 = 420.
The sequence begins:
1;
2;
4, 6;
8, 12, 30;
16, 24, 36, 60, 210;
32, 48, 72, 120, 180, 420, 2310;
64, 96, 144, 240, 216, 360, 840, 900, 1260, 4620, 30030;
...
MAPLE
with(combinat): A063008_row := proc(n) local e, w, r;
r := proc(L) local B, i; B := NULL;
for i from nops(L) by -1 to 1 do
B := B, L[i] od; [%] end:
w := proc(e) local i, m, p, P; m := infinity;
P := permute([seq(ithprime(i), i=1..nops(e))]);
for p in P do m := min(m, mul(p[i]^e[i], i=1..nops(e))) od end:
[seq(w(e), e = r(partition(n)))] end:
seq(print(A063008_row(i)), i=0..6); # Peter Luschny, Jan 23 2011
# second Maple program:
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-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n$2))[]:
seq(T(n), n=0..9); # Alois P. Heinz, Sep 03 2019
MATHEMATICA
row[n_] := Product[ Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 10 2012 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[ Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001
EXTENSIONS
Partially edited by N. J. A. Sloane, May 15, at the suggestion of R. J. Mathar
Corrected and (minor) edited by Daniel Forgues, Jan 03 2011
STATUS
approved