A063008 Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,...] with p1 >= p2 >= p3 >= ... is encoded as 2^p1 * 3^p2 * 5^p3 * ... .

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, 128, 192, 288, 480, 432, 720, 1680, 1080, 1800, 2520, 9240, 6300, 13860, 60060, 510510, 256, 384, 576, 960, 864, 1440, 3360

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

Cf. A001222 (bigomega), A025487, A059901.

See A080576 Maple (graded reflected lexicographic) ordering.

See A080577 Mathematica (graded reverse lexicographic) ordering.

See A036036 "Abramowitz and Stegun" (graded reflected colexicographic) ordering.

See A036037 for graded colexicographic ordering.

Sequence in context: A087443 A059901 A036035 * A303555 A136101 A187779

Adjacent sequences: A063005 A063006 A063007 * A063009 A063010 A063011

Keyword

nonn,look,tabf

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